The honest answer: almost nothing does true two-players-on-one-phone
If you are two people with one phone between you, the setup to use is pass-and-play - a turn-based game where you hand the handset over between turns - and the best pick for it is Monopoly, which supports up to six players locally sharing one device or controller.
But it is worth being straight about why this search is frustrating: simultaneous two-player action on a single phone screen barely exists as a category. A phone has one small screen and no room for two people's thumbs without them colliding. A handful of games have tried a split-down-the-middle layout over the years, but it is a novelty format rather than a supported one, and most of the lists promising it are padded with games that do not actually have it. If you came here expecting a long list, that is the reason there is not one.
The good news is that the underlying problem - two people, one device, nothing else available - has three real solutions, and two of them are better than the thing you were searching for.
Setup 1 - Pass-and-play, the one that genuinely needs one device
Turn-based games do not need two sets of thumbs at once, so one device is enough by design. You take your turn, you hand it over. Board game adaptations are the natural home for this, and Monopoly is the pick: it supports up to six players locally with one controller or device passed around, which is exactly the shape of this problem.
This is the only setup on this page that truly needs nothing but the one device you already have in your hand. Its limit is pace - it suits a sit-down evening, not ten minutes of something fast.
Setup 2 - One screen, phones as controllers (the inversion)
This is the setup most people should actually use, and it solves the problem backwards. Instead of two people sharing one small screen, everybody's phone becomes a controller and the game runs on a television or a laptop.
The Jackbox Party Packgames work this way: one person owns and runs the game, and everyone else joins from their own phone's browser with a room code. Nobody else buys anything, nobody installs anything, and it scales to a whole room rather than two people. If you have any screen in the house bigger than a phone, this beats sharing a handset outright.
Setup 3 - Two phones and a free crossplay game
Worth stating because it is often the real situation: people search for "two player games on one phone" when what they have is two phones and no money. That is a much easier problem. Among Us and Fall Guys are both free, both run on phones, and both crossplay, so two people on two handsets can be playing in a few minutes at no cost.
Among Us is the better of the two here, because it also works over a call and supports a much larger group if more people join later. The full list is at free mobile games and free crossplay games.
What to check before you commit an evening
Check for "local" or "pass and play" in the store listing, not "multiplayer".On mobile, "multiplayer" almost always means online with strangers. Local same-device play is a separate feature and is listed separately when it exists.
Beware the hot-seat trap in free-to-play games. Some mobile games tie progress to a single account, so handing the phone over means your friend is playing your save. Fine for a board game, annoying in anything with progression.
Battery and heat decide the session length. One phone doing all the work for two people, often while being passed around and held, gets warm and drains fast. Keep it on a charger if the evening is going to run.
If the two of you are not in the same room, this is the wrong page. Two people apart with phones want games over a phone call or games over FaceTime, both of which assume a device each.