They will either reuse the address in coinbase transaction subsidy output every time they mine a block, and then publicly claim that address, or they will "tag" their name (ascii-encoded) and some metadata in an OPRETURN or the scriptSig of the coinbase transaction.
Here's the coinbase TX from block 603526:
https://blockstream.info/tx/0eb39c3f1a6813dfe5bb9d86dd71de2593aa76f25d4f25b149c9f564eb33cd54
The script sig is
03863509fabe6d6d0f3e894118566fd52beccf2a405c060fb701171dd16d83f1fb4aa0219fa8656601000000000000002469020000535e15a74a1b00000000e6892f736c7573682f
which is ASCII for:
5 z>mm> AVoU+lO*@\7Qmq{J !(ef$iS^'J /slush/
There's slushpool!