Python 2, 20 bytes

lambda*a:map(max,*a)

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I/O as 0-pre-padded lists of digits.

Jelly, 1 byte

»

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I/O as 0-pre-padded lists of digits.

R, 68 65 bytes

function(x)apply(outer(x,10^(max(nchar(x)):1-1),`%/%`)%%10,2,max)

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Input as integers, output as list of digits.

If zero-padding lists of digits was allowed, then simply pmax would suffice.

MATL, 2 bytes

X>

Choose the most appropriate format for your language/solution

The input format is: 2D char array of two rows, each corresponding to a line, with the shorter number left-padded with spaces. For example

17210

  701

which in MATL is defined as

['17210'; '  701']

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Explanation

      % Implicit input: 2D char array with two rows 

X>    % Take maximum of (code points of) each column

      % Implicit display

Python 2, 73 60 56 bytes

lambda a,b:map(max,zip(a.rjust(len(b)),b.rjust(len(a))))

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Takes input as two strings, and returns a list of digits

Alternative:

Takes input as two integers; same output

Python 2, 60 59 bytes

lambda*i:map(max,zip(*['%*d'%(len(`max(i)`),v)for v in i]))

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J, 14 12 bytes

-2 bytes thanks to Jonah

(>./@,:)&.|.

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Input and output as list(s) of digits

Japt, 9 8 7 bytes

Takes input as an array of digit arrays.

mÔÕÔËrw

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m            :Map

 Ô           :  Reverse

  Õ          :Transpose

   Ô         :Reverse

    Ë        :Map

     r       :  Reduce by

      w      :  Maximum

If taking zero-padded arrays as input is permitted (it would currently fall under a "convenient format" but I suspect that's not the challenger's intent) then this can be 3 bytes.

íwV

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í       :Interleave the first input

  V     :With the second

 w      :Reduce each pair by maximum

Perl 6, 37 bytes

{[R~] roundrobin($_».reverse)».max}

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Takes input as a list of lists of digits.

05AB1E, 9 6 5 bytes

-3 thanks to Emigna

-1 thanks to Shaggy

íζ€àR

Takes input as a list of lists of digits

í      # Reverse both inputs

 ζ     # Zip

  ۈ   # Keep the bigger digits

    R  # Reverse

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Java 10, 78 57 bytes

a->b->{for(int i=a.length;i-->0;)if(a[i]<b[i])a[i]=b[i];}

Input as two space-padded character arrays.

Modifies the first input-array instead of returning a new one to save 21 bytes (thanks to @OlivierGrégoire).

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Explanation:

a->b->{            // Method with two char-array parameters and String return-type

  for(int i=a.length;i-->0;)

                   //  Loop `i` in the range (length, 0]:

    if(a[i]<b[i])  //   If the `i`'th character in input `a` is smaller than in input `b`:

      a[i]=b[i];}  //    Change the `i`'th character in `a` to the `i`'th character of `b`

Haskell, 40 bytes

a#b=zipWith max(p b++a)$p a++b

p=(' '<$)

Input/output as strings, try it online!

Explanation

The function p replaces each character by a space, using p b++a and p a++b are thus the same length. This way we can use zipWith without losing any elements, using max with it works because a (space) has lower codepoint than any of the characters ['0'..'9'].

JavaScript (ES6), 51 49 bytes

NB: This answer was posted before the loose I/O formats were explicitly allowed. With zero-padded arrays of digits, this can be done in 33 bytes, (but is much less interesting, IMHO).

Takes input as two integers. Returns an integer.

f=(a,b,t=10)=>a|b&&(a%t<b%t?b:a)%t+t*f(a/t,b/t)|0

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Commented

f = (                     // f = recursive function taking:

  a,                      //   a = first integer

  b,                      //   b = second integer

  t = 10                  //   t = 10 (which is used 6 times below)

) =>                      //

  a | b                   // bitwise OR between a and b to test whether at least one of

                          // them still has an integer part

  &&                      // if not, stop recursion; otherwise:

  (                       //

    a % t < b % t ? b : a // if a % 10 is less than b % 10: use b; otherwise: use a

  ) % t +                 // isolate the last decimal digit of the selected number

  t *                     // add 10 times the result of

  f(a / t, b / t)         // a recursive call with a / 10 and b / 10

  | 0                     // bitwise OR with 0 to isolate the integer part

Alternate version

Same I/O format.

f=(a,b)=>a|b&&[f(a/10,b/10)]+(a%10<b%10?b:a)%10|0

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Perl 6, 15 bytes

{[~] [Zmax] $_}

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Takes input as a list of space padded arrays of characters, though for this challenge the lax input format makes it rather boring. Alternatively, here's the program that takes a list of two integers instead:

Perl 6, 41 bytes

{+[~] [Zmax] $_>>.fmt("%{.max}d")>>.comb}

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If you don't mind a huge amount of whitespace, you can also remove the + from the front.

Explanation:

{                                       }  # Anonymous code block

             $_>>    # Map each integer to 

                 .fmt("%{.max}d") # The number padded by the max of the list spaces

                                 >>.comb   # And split each to list of characters

      [Zmax]  # Get the max of each digit at each index

              # This works because space is coerced to 0

              # Otherwise we would have to add a 0 to the formatting string

  [~]   # Join the list of digits and spaces

 +      # And coerce the string to a number to get rid of leading whitespace

Tcl, 156 bytes

proc S a b {join [lmap x [split [format %0[set l [expr max([string le $a],[string le $b])]]d $a] ""] y [split [format %0$ld $b] ""] {expr max($x,$y)}] ""}

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Not very golfy, but I had to give a try on it. Will golf it later!

Batch, 120 bytes

@set/aw=1,x=%1,y=%2,z=0

@for /l %%i in (0,1,9)do @set/a"z+=w*((v=y%%10)+(v-=x%%10)*(v>>4)),y/=10,x/=10,w*=10

@echo %z%

Takes input as command-line parameters. 188-byte version works on arbitrary length integers:

@set/px=

@set/py=

@set z=

:l

@if %x:~-1% gtr %y:~-1% (set z=%x:~-1%%z%)else set z=%y:~-1%%z%

@set x=%x:~,-1%

@set y=%y:~,-1%

@if "%x%" neq "" if "%y%" neq "" goto l

@echo %x%%y%%z%

Takes input on STDIN.

Husk, 5 bytes

↔¤żY↔

Conveniently takes input/output as list of digits, try it online or verify all!

Explanation

↔¤żY↔  -- example inputs [1,4] [3,2]

 ¤  ↔  -- reverse the arguments of: [4,1] [2,3]

  żY   -- | zipWith (keeping elements of longer) max: [4,3]

↔      -- reverse: [3,4]

Stax, 5 bytes

|>E:o

Run and debug it

This program takes input as an array of strings.

|>  Right align inputs (filling with )

E   "Explode" array onto stack separately

:o  "Overlay" Keep the maximum element respective element from two arrays.

Run this one

This is the first time I've seen a use for the overlay instruction "in the wild".

SNOBOL4 (CSNOBOL4), 153 bytes

	X =INPUT

	Y =INPUT

	Y =DUPL(0,SIZE(X) - SIZE(Y)) Y

S	X LEN(1) . A REM . X	:F(O)

	Y LEN(1) . B REM . Y

	O =O GT(A,B) A	:S(S)

	O =O B	:(S)

O	OUTPUT =O

END

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Retina, 45 bytes

P^`.+

+`^(.*)(.)¶(.*)(.)

$1¶$3¶$2$4

%O`.

¶.?

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P^`.+

Pad both values to the same length.

+`^(.*)(.)¶(.*)(.)

$1¶$3¶$2$4

Transpose the values.

%O`.

Sort each pair into order.

¶.?

Delete all the low digits and surplus newlines.

Charcoal, 14 bytes

⭆◧θＬη⌈⟦ι§◧ηＬθκ

Try it online! Link is to verbose version of code. Explanation:

  θ             First input

 ◧              Padded to

   Ｌ            Length of

    η           Second input

⭆               Map over characters and join

     ⌈          Maximum of

      ⟦         List of

       ι        Current character of first input and

          η     Second input

         ◧      Padded to

           Ｌ    Length of

            θ   First input

        §       Indexed by

             κ  Current index

                Implicitly print

17-byte version "adds" any number of strings:

≔⌈ＥθＬιη⭆η⌈Ｅθ§◧ληκ

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Pyth, 5 bytes

meSdC

Takes input as array of two space-padded strings.

meSd       map greatest

    C      on the transpose of input

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Twig, 125 bytes

When I saw this challenge, I though: "let me use a template language! sure is a good fit"

I was wrong ... so wrong .... ... But was fun!

{%macro a(a,b,s='')%}{%for k,x in a|reverse|split('')%}{%set s=max(x,(b|reverse|split('')[k]))~s%}{%endfor%}{{s}}{%endmacro%}

This requires that "strict_variables" is set to false (default value).

To use this macro, you can do like this:

{% import 'file.twig' as my_macro %}



{{ my_macro.a(195,67) }}

Should display 167.

You can try this in https://twigfiddle.com/rg0biy

("strict_variables" set to off, it is on by default on the website)

C# (.NET Core), 58 bytes

(a,b)=>a.Select((x,i)=>a[i]>b[i]?x:b[i])

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Japt, 4 bytes

Input is taken as a an array of two 0-padded number arrays.

y_rw

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Mathematica 50 bytes

a = 543;
b = 791;

FromDigits@MapThread[Max, IntegerDigits /@ {a, b}]

(* 793 *)