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So based on my limited understanding, some luxury cars have taken to using their sound system to make the cabin quieter when on the highway or something (Active noise control via wikipedia). And I learned about destructive wave interference in high school physics.

First of all, this is a thing, right? Secondly, I know vaguely how suppressors work and they're not strictly "silencers", but would using some sort of sound system with "anti-noise" be possible for reducing weapon report? And if it's not feasible with technology or whatever IRL, without worrying about such limitations, would it make sense in an advanced society?

If the gun is suppressed and has subsonic muzzle velocity, then noise cancellation should work for the remaining noise. In practice I am not sure if it would be worth it as a suppressed gun firing subsonic ammunition is usually good enough especially if you use a captive piston ammunition. But you'd only need it at few locations so the cost might be low enough for the reduction in noise to be worth it.

The reason gun needs to be suppressed and subsonic first is because otherwise the gun will emit noise that noise cancellation is not really feasible against. You cannot really do anything about the shock waves caused by supersonic projectiles or propellants for example without raising questions why you are using a bullet at all instead of just weaponizing your technology to kill people without physical bullets. Same with negating noise from the propellant in general. If your electric system can negate it practically, you might as well skip the propellant and use your technology to directly drive the bullet without using chemical propellant at all.

So my answer would be that using it as a replacement for current "silencing" technologies would not make sense but it might be used in addition to them for minor gains.

Noise cancellation is a tricky thing.

Noise cancelling uses a soundwave with an opposite "wave" to cancel the other out. But since its a wave you essentially create an expanding bubble from the center of the Noise. Since you cant have your speaker perfecrly on top of the center of the Noise you'll create an interference pattern: at some places the Noise is cancelled, at some the Noise is doubled!!

Noise cancelling works by using a sound you already know is going to be made and a range of speakers placed inside the room. For a project we could only cancel out repetitive noises that could be predicted. We also needed to know the way sound bounced through the room (far beyond my expertise) and where people would be to make sure the double-noise area's happened somewhere people wouldnt be. So if your Noise cancelling worked it would need to know the placement of every person upon firing your weapon and the environment you are in at the same time. Worse: with speakers on the muzzle you might stop the muzzle sound but not the sound of the bullet breaking the sound barrier.

I dont think Noise cancellation will work very well to silence weapons.