A few NFT collections have sold or auctioned 'mint passes,' which later let holders mint an NFT from the actual collection.
Are these mint passes just regular NFTs, and the next NFT contract checks that the wallet is holding one before allowing them to mint a new NFT?
This would depend on the contract logic. You might be able to find the actual code of the smart contract behind the mint passes, on github or Etherscan, so you can clarify that directly.
There are multiple options available how to implement that, starting from regular NFTs or some capped issuance fungible tokens such as ERC-20. It could also be a data structure (mapping) in your smart contract, it could be ERC-1155 structure etc.
In one of our implementations we have done initial allocation (Sale/Reservation) via ERC-1155 smart contract and blocked transfers in there until the claim event, while the actual final issuance (Claim) is done via ERC-721 which reads data from ERC-1155.
