It would be nice to have the ability to silence people's messages and game invites while you're playing a game, like uno for example.

There is definitely room for a feature like this, and I'm glad I'm not the only one bothered by the same thing. Please add this for our sanity.

Hello,
I am just thinking about one problem here.
For example, sticking specifically with Uno, if you were to enable such a feature, how will you know when a person tells you to wait because they have to temporarily go away?

Other than that, if the sounds are the only thing that bothers you, temporarily you can lower the volume of notifications to 1 percent while you are playing.
If you want to do that though, I'd first go to table messages in options/reading and appearance/channels and turn them from notifications to normal sounds, as if you don't do that you also won't hear when people leave/join your table.

My bad, I don't think I made myself too clear. This would specifically be for private messages and invites. Table chat messages are not as loud as those ones and as you said, those are sometimes necessary for knowing when to pause the game. The ones who aren't apart of the game or the table should simply shut up, but saying that to them isn't too polite. An idea would be to have a mute toggle button that can be assigned certain views and notifications.

How about the ability to create and store custom configurations? I'm imagining something like NVDA profiles or Voiceover activities where you can create a custom config from reading and appearances settings, and switch to them upon joining a table or via a shortcut key.