My wife and I own two PC's, one hooked up to our TV and one on a monitor beside it. All games have been played on separate PC's, so none of our recommendations are based on how well the devs implemented split screens. Two PC's (if you can afford it) is by far the way to go if you play with someone regularly.
I have a much broader interest in games than my wife does. My wife gets overwhelmed and motion sick with many first person games, so we mostly stick with third person games. She also doesn't like competitive online games and so most of the game we play are player vs. environment.
This was absolutely fantastic. It's a little bit of a slower game since it's turned based, but there's a lot of depth and a lot of opportunity to work together.
Divinity: Original Sin 2 - Definitive Edition on Steam
This game is SO much fun in co-op, I can't believe that it's a mod and not an official part of the game. We regularly play together as it's easy to play a single run here and there. We use this co-op mod . There's also another one that many like, but we didn't as much. Each player needs to take turns (or you race each other)
A bit like if Slay the Spire was designed from the ground up for Co-Op. Fantastic, addicting game!
Skyrim co-op was made possible by the Skyrim Together mod. We had a blast playing this together. My wife prefers more action than Skyrim has though, so was a little bored with world-discovery.
The Elder Scrolls V: Skyrim Special Edition on Steam