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It's google. They know that you downloaded the app from the Play Store and used the app to connect to their servers directly. They already have plenty of information.

Yes, showing you an advertisement for an app you've already installed is bad UI/UX, regardless of the reason why. On the publisher side, it's also a wasted ad impression.

I don't expect them to know all the apps installed on my phone, nor do I think they need that much information to solve this particular problem.

If they don't know it, they shouldn't make assumptions.