Against the Storm (and excellent rouguelite city-builder) does this in a really cool way. Pausing is a core mechanic of the game, and you frequently pause while you place building or things like that - and all the visual animations stop (fire, rain, trees swaying, etc).

But when you find a broken ancient seal in the forest, the giant creepy eyeball moving around in it keeps moving even when you pause the game, which helps emphasise how other-worldly it is.

I find it confusing: for me a clear indicator the game is paused is all animations also pausing. Some games do not pause in menu’s, for example. And some do, but not when in a multiplayer session

I'm the opposite, it drives me crazy! Along with sound effects / music playing.

Torch flames and trees swaying in the wind do not affect gameplay at all - they're most likely done in a shader and I think it's easier to keep updating a time uniform than to add extra conditions everywhere :D

That's usually because the system that runs those things is independent of the timing of the main game loop. And then when someone finally gets around to implementing the pause screen, they still run even with the main game time stopped. And you look at it and think "eh, you know what - looks cool - we'll leave it".

It's nice when a bug manifests as a feature.