I doubt it. That would require all those features to exist in the first place. Unlike Steam you do not need the client, neither to download games, nor to play them. You can do all on the website and most games have no drm, so there is no need to fire up the itch app to run them.
There might be planned features. Best to check out the faq about the itch app. There seem to be some apis for online games, but it reads like you need your own server for that one.
I imagine browser games would be the first to get any integration. They can already be cached by the itch app in some way. At least playing them with the app counts as a download, as I read somewhere in the forum.