They are not "coded into the ethereum blockchain". The media file is still uploaded on either a centralized server, or through IPFS. The only thing that goes into the blockchain(unless you're using something like Arweave) is the digital signature.
So a painting, music, video game NFTs are all uploaded to the server but the blockchain is only there to confirm authenticity of product via digital signature?
Not all NFTs come with digital signatures from the creator. NFT at its core is just a unique token, and this technology isn't even new. Like others have mentioned, there are different ways of establishing the link between the token and the thing that it is supposed to represent, and most of the time such link is quite weak.
NFTs are not officially recognized by any government, which can often mean that they are practically worthless, because you can't defend your ownership if it would get violated.