The same mnemonic generates two addresses on the test network:
A mnemonic is a seed to generate private keys not addresses. The address is later derived from the public keys derived from those private keys by the wallet you are using and can be of any type.
Can I mine test coins? I have some mining machines.
You can, but since the block height is too high (2.5 million blocks) the block reward at this point is very small at about 0.012 (TestNet has had far more halvings than MainNet).
Just receive the coins from faucet into a different address then send them to your taproot address.
Or you could just use RegTest.