Bitcoin has much wider adopt which is most likely why it is first. BTC has an average of 600k transactions per day vs XMR which averages about 30k per day. I’d probably make the one that would actually be used the most first as well.
And you’re complaining about this while using the free tier of a private service that doesn’t sell your data? If I were on the free tier I’d trade a signature and blockable pop-ups (uBO) for not being the product.
If you like syncing across devices, Bitwarden (paid) or Proton Pass. If you already use Bitwarden and don’t like the idea of having passwords and 2FA in the same place, just make a second Bitwarden account.
It’s most likely an API linked to your logged in account that counts each search performed and logs it against your account. You don’t need to see a user’s search to track that information.
Like it or not, it still has a larger user base.