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There are at least two U2F keys I was able to find on Amazon which were under $10 USD:

https://www.amazon.com/dp/B01N6XNC01 https://www.amazon.com/dp/B01L9DUPK6

The latter is an open source / open hardware design:

Neither of them works with smartphone hence not of practical use.

I think you might be having trouble with the concept of a U2F token. The mail application on your phone doesn't need a token; it already has a long-term cryptographic binding to the server.

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