I don’t think a site like yours will be able to have access to the Drops system, as it’s not a game but a 3rd party service. Drops are only available when a streamer sets the game they are playing to one with access to Drops, which would mean even if you did somehow manage to get your site registered as a ‘game’, the streamer would actually have to set that as the game they are playing and that be the focus of the stream, rather than properly streaming.
Also, I’m curious as to why you want to have access to the drops system in the first place? It’s not something that’s aimed towards 3rd party services like yours, and there are sites that have done what you plan to do using what’s already available to them through the API and not needing Drops.