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Two players, one phone
The most social thing a browser game can do is fit two people around one screen.
Online multiplayer is expensive to build and easy to get wrong. Two people on one device is neither. It costs the publisher nothing and it turns a solo game into a small event at a table.
SowSoy collects these games on the Co-op counter, using only the games whose publisher explicitly lists a two-player mode. It is a short list, and that is honest: most browser games are built for one person.
The ones that qualify tend to be turn-based or split by timing — a board, a race, a pair of controllers on a single keyboard. They are the games that survive a cousin visiting, a long wait at a clinic, or a rainy afternoon with nothing charged.
If you are choosing for two people, look for games with simple rules and a visible score. Competitive games work better than collaborative ones on a shared screen, because the argument about the next move is the point.
And if only one of you wants to play, the Co-op counter is still useful: a two-player tag often means the game is not punishing, which makes it a good evening pick for a single person too.