What is being asked is to know who is playing your game if this person is logged in, and that's all. This information is public (if the person decides to publish something).
What do you think public means in this context?
The information who is playing which games is not public. It is not even known to the dev that published the web game. Hence this very thread.
Even if the existing api is capable of doing this, when playing a game or a web game with the Itch app, using the username inside the game (like displaying it on a leaderboard) would violate some privacy boundaries. Itch is just not a place like those web game hosters with the microtransactions or facebook games or even Steam.
Itch might one day have such capabilities, but as I said elsewhere here, there would need to happen a major overhaul including a huge bunch of other multiplayer support features that enable and control the interaction between users while playing online together.