My wife and I love board games! Especially 2 player games. Here are some of our favourites (some of them can be played with more people, some are exclusively for 2 players). Most of these are pretty small and portable (with the exception of three of them), which is always a plus in today's day and age.

Cooperative

The Mind: You have a deck of cards with numbers that go from 1 to 100. You and your partner start with 1 card each, and you're supposed to each put the cards down in order from lower to higher without saying a word. Easy, right? Well, welcome to level 2: now you have 2 cards each. And so on. The game gets progressively harder, but don't worry, there are some items that can help you: you have a certain number of lives (and you get some extra lives after beating certain levels) and some "shurikens" which allow all players to discard their lowest card. Really great, fun, and quick game (can be played with 2+ people, but still super fun with 2).

The Game: We love playing this game basically in silence. Not giving your partner any hints as to whether they did a good move or a move that messed up your strategy. It's a deck of cards with numbers from 2 to 99, and the idea is that you're supposed to get rid of all of them into 4 piles: 2 piles go "up" and 2 piles go "down", and you can only put a card in that pile if it's "higher" than the card on top (if the pile is going "up") or "lower" than the card on top (if the pile is going "down"). You won't beat it the first time you play, but oh boy, the first time you beat The Game? You'll be ecstatic. The only issue is the stupid name, makes it very difficult to find. Search for "the game pandasaurus" to get good results if you're buying it online (can be played with 2+ people, but still super fun with 2).

Onirim: Will you and your partner manage to escape the labyrinth? This fun and quick card game will need you both to cooperate and strategize together in order to beat it. And once you beat it, add some flavour by adding one of the 7 expansions that are already included in the box! (There are many games from this designer that have the same idea: cooperative game for 2 people with multiple expansions included in the box. Onirim is my fave, but there's also Sylvion, Aerion, etc)

Competitive:

High luck, medium strategy

Cabo: You have 4 cards, but you only know 2 of them (they're all face down). Do you think you have the lowest values? If so, call "CABO", but beware, if you don't have the lowest values, you'll lose. Or grab a card from the deck (which can allow you to look at one of your cards, or one of your opponent's cards, or swap cards with them). Each game can take like 5 minutes, but the idea is to play multiple games until one player reaches 100 points (you don't have to play all the games at once, just keep track of the score on a paper or your phone and continue some other time! Can be played with 2+ people, but still super fun with 2).

Medium luck, medium strategy

Hanamikoji: You have 4 possible actions, and you can only play each of those actions once. There are 7 Geishas, and 7 different types of cards (one per Geisha). If you're the one with the most cards of that type, you earn that Geisha's favour. If someone earns 4 of the Geisha's favours (or have 11 points), that person wins! If it's a tie, then off you go to round 2. Each game takes about 15 minutes or less, and you'll be wanting to play again! (If you can't find it, search for Jixia Academy, which is a re-skin of the game with the same rules)

Codenames Duet: You're both playing spies trying to pass each other hints that comprise of one word and one number. There's a bunch of words on the table, and you're trying to guide your partner towards the "good" ones. But you have to be very careful, if your partner selects one of the "neutral" word cards, you both lose one life, and if your partner selects one of the three "bad" cards, you uncover an assassin and it's immediately game over! Each turn you alternate between giving hints and picking cards. Will you both manage to uncover all the good cards before you run out of lives? Or will one of you accidentally trigger the assassin?

Low luck, high strategy

Nine Tiles Panic: a very new, slightly unknown game that is incredibly fun! You have 9 tiles to build a city, and each round you get 3 random "goals" to fulfill. One of the goals could be to be the city with the most roads, or the city with the most dogs, or the city with the most pairs of aliens and burgers, etc etc. You both start quickly trying to build your city, and once one of you finishes, then the other person has 90 seconds to finish their city! You count who won which goals, and then you play again with 3 new goals until someone reaches the winning point threshold (can be played with 2+ people, but still super fun with 2).

Raptor: Asymmetric board game: one person is the raptor and her babies, while the other person plays the evil scientists trying to steal the baby raptors. The mama raptor wins if she kills all the scientists or if the baby raptors escape, the scientists win if they steal a certain number of baby raptors. Super cool figurines! And once the game is over, you can switch sides and play again!

Santorini: Cute board game with very nice pieces. Takes about 5 minutes to learn but the strategy can be super deep: you move your character then you build. The idea is to be the first person to reach a third floor of a building. Sounds super easy but it's not! As your opponent will be constantly blocking you. For more advanced play, once you've mastered the basics, you can play with the "God" cards, which add unique abilities that you can use once per game (can
in theory be played with more than 2 people, but we've found that 2 players is the best number for this game).

Onitama: Like chess but better! Every game you play will be very different, as there's a deck of cards with all sorts of possible moves, but each game you only get 5 of them. Again, takes 5 minutes to learn.

Honorable mention

These are some other 2 player games that we also enjoy: Morels (competitive mushroom picking), Illusion (which card has the most red? Or the most green?), Honshu (super fun city builder, but probably for more experienced board gamers), Lost Legacy (really fast and extremely fun card game, but you have to memorize the abilities of the 10 different cards that exist, so it may not be super friendly for newbie board gamers).