Recently there was a thing where VAC would erroneously flag AMD’s antilag+ feature as cheating, and issue a ban.
AMD then quickly disabled the feature by default but now Valve also patched detection for it and is now, at least according to these patch notes, reversing the bans.
If there is a VAC ban on your account that people can look up servers on other games might ban you as well.
Sure, there’s a possibility. But since it hasn’t happened in 10+ years I doubt it’s a problem.
My friend can’t play Hell Let Loose because of a game ban on a completely different game. Different anticheat, and different publisher. So. I would be shocked if a VAC ban wouldn’t cause the same issues.