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?
The media file may or may not be uploaded to any server, but I think that uploading to IPFS is common. The NFT itself lives in a blockchain. An NFT generally contains a hash and description of the media file that associates the NFT with the media file.
Note that ownership of digital art is handled by copyright law, and owning an NFT does not necessarily grant you ownership of the copyright. Similarly, owning a physical piece of art does not automatically grant you ownership of the copyright. Some artists grant ownership of the copyright to the owner of the NFT, but others do not.