It is possible. Such manufacturer could create HD wallet and publish its public key. Then, each device could have one of the child private keys. In this way, any miner using such equipment could use it to sign any message and prove that it has such equipment. For me, it seems pointless, but technically it is possible.
When solo mining, proving that particular equipment was used to mine a block is trivial, all that is needed is checking the coinbase transaction. When mining in pools, proving that is harder, because it is possible if miners will switch to Stratum v2 or similar protocols where they can control their blocks.
True! Ignoring some of the other practical issues (e.g. pool mining vs solo mining) this brings up another interesting problem though:
1) Assuming non-hardened child private keys, once the private key of a single miner is compromised, the private keys of
all miners are compromised [1]. Keeping private keys secure from someone who has full physical access is somewhat possible, but quite a challenge. Especially if the pot is not the contents a single hardware wallet but a whole network of mining hardware.
2) Assuming hardened child private keys, you get rid of the security concern above, but the manufacturer would not be able to verify the signatures [2]. Unless they derive and store each hardened private key in advance which comes with a different host of issues.
[1]
https://bitcoin.stackexchange.com/questions/90627/what-are-the-consequences-from-the-leak-of-xpub-and-child-private-key[2]
https://learnmeabitcoin.com/technical/extended-keys