No — the Chrome dino game has no multiplayer mode of any kind. There's no versus screen, no online lobby, and no co-op. Here's why, and what the "multiplayer dino" results you may have seen actually are.
Why it was built single-player
The dino was designed in 2014 as a tiny time-filler for Chrome's offline error page. By definition you had no connection when you saw it — so online multiplayer was never an option, and the team kept it to one input, one player. More on its origins in who made the dino game.
What "multiplayer dino" sites really are
Third-party remakes sometimes bolt a ghost-race or shared-leaderboard layer onto a dino-style runner. Those are fan projects, not Google's game — treat them like any unofficial site, and know that your Chrome high score never syncs anywhere (it lives only in your browser).
Competing without a multiplayer mode
Score-chasing against friends works brilliantly anyway — turn-taking, same-start races, best-of-five. We've written a full set of formats with fairness rules in dino game 2 player: make it a contest.