My favorite game in 2001 was medieval total war. I love turn based nation building strategy games where being a joystick legend is not an issue. Played a lot of final fantasy in the 90's. I am having difficulty finding pure strategy games that actually have a following (probably a sign of this generation). I am courting the idea of Europa Universalis, but, I am not sure. My son and I enjoy chess together, although, I have to say it gets boring.
I am a single player at heart, so not sure how much help I can be.
Total War series has a lot of newer titles, so maybe one of them? I mean there is a Medieval Total War II for example - though you probably already mean that, not sure when it was out ;)
If you are into villages builders, "Medieval Dynasty" has a MP option now. No strategy though, but I would gues good for bonding. Unfortunately SP only is "Songs of Syx", an indie perl.
The Paradox game most are into is Stellaris, though I heard it's not that stable in MP (at least 2 years ago). As the name suggests, SciFi Grand Strategy.
And for anyone that plays chess, I always try to proselytising them to
Go/Weichi/Baduk! Go is way better! More strategy, less confusion!
If "Battlemechs" say anything to you, there is a wonderful turn-based (though it's mostly intended as a SP campaign game, you can play MP fights) strategy game "Battletech".
If you like to build your own spaceships with lots of weapons on them (not much strategy though), there is Avorion.
"Shieldwall" is funny little game (though slighly comic-blood bloody) too and I would be surprised if not MP. No strategy, only nasty tactics though.
Master of Magic has also gotten a remake (and the old one out-of-the-box patched steam version for a few bucks). If you somehow don't know this game even you are an old "turn based nation building strategy games" person: This is basically Civilization with Magic. Enchanting! I actually can't say how good the remake is, the votes are not that good. (The same company also failed at Alpha Centauri remake, but did others well.)
For FPS - if you have fun with medieval settings, there is no avoiding
Mount & Blade II Bannerlord. You can also build your own kingdom, but that is SP campaign only Out of the boy that is: there is a co-op mod. Can't say anything about it, except that people seem to be very happy:
https://www.youtube.com/results?search_query=Bannerlord+online+is+mindblowingGenerally you can also have a look at
"Hooded Horse" publisher. "Old World" is similar to Civ. The rest might not be in your primary target, but all the games seem to be in their own special niche and doing that one very good. I certainly have them on my watch list.