First, Mastodon isn’t a platform, it’s a service. Unlike Mastodon, Android was always a bunch of proprietary stuff built onto an open source base. The Android license (Apache) is also a lot more permissive than Mastodon’s (GPL). Probably the most important thing here is that all derivative works must be licensed under the GPL, whereas Google can use AOSP code to build out proprietary features whenever they want.

Their ability to use the app to direct users to mastodon.social depends entirely on Mastodon’s good reputation. Destroying the reputation destroys the ability along with it. Mastodon is way bigger than just m.s, but a buyer wouldn’t control the instance in a meaningful enough sense. Users aren’t serfs and there would be a mass exodus if, say, Peter Thiel bought Mastodon. Some would stay, but the people who contribute probably 90% of the activity would be out the door. Very likely, users would be given time to migrate before the larger community defederated the instance en masse. Any effort to prevent users from leaving would just accelerate that process. They just have no real ability to compel people to behave the way they want.

To be fair, mastodon.social is Mastodon instance, and all Mastodon instances (that support signup at all) will show a Sign Up link for themselves. This is true from the biggest instances to the smallest ones.

But to bolster your point, the Mastodon app could push exclusively for signups on the newly acquired mastodon.social domain, and hide other options behind a dark pattern… Like how Element does this when you try to log in to a Matrix server.