Cryptopunks are way older than eip721. Now this eip is accepted as the standard but punks were not able to comply to it as it did not exist in 2017.
It's not an issue, it only means that marketplaces have to specifically know how to interact with this Punks contract instead of relying on the standard, but it's still a totally real NFT.
The reason why you should follow the standard is that it will make your NFT easily tradeable on markets like OpenSea and be able to interact with games and all places where NFTs are going to be used (the famous metaverse).
But if you want to create your own NFT contract and let people use it only from a specific dapps (yours) or interact directly with the contract by knowing the ABI (publish source code in Etherscan), you can do it and it will still be called an NFT. Just don't expect much traction as it will not be easy to use. But it may be your goal, for instance if you only target experts.
Interestingly you can comply with eips and be creative in the implementation. It's up to you.