I think it really just depends on the game and what purposes “pausing” serves in that game. Take a game like solitaire, for example: there is no meaningful “pause” feature you could add, since the game state only advances in response to a user action.

Other pause some underlying simulation while still letting you modify the game state, as an expected part of gameplay, like a city builder. As the user might spend a significant amount of time in a paused state building things, it would be pretty visually unappealing to have the entire world completely frozen the whole time.

Others might pause all gameplay entirely, such as for displaying a menu, in which case pausing even environmental animations might make more sense since the user isn’t actively playing.

For the second type, I would much prefer some GUI element to indicate the simulation is paused rather than freezing the whole game world, such as a border around the screen or maybe a change of color theme of the GUI or similar.