It's not a wallet in the conventional sense, whether it's Metamask, Electrum, MEW, Trust, Coinomi, Atomic, or any other program you consider a wallet.
Just a comment here: you should not be using any of that wallet software except Electrum. They have been suffering from security vulnerabilities and Atomic wallet even got hacked and people lost millions with no rationale of what happened yet;
https://bitcointalk.org/index.php?topic=5455045.0Your private key or seed phrase is what makes up your wallet.
Technically speaking, a wallet is consisted of the transaction labels as well, none of which is backed by a seed phrase.
Blockchain technology is so revolutionary because of this.
Blockchain technology is mainly innovative because of the immutability it offers. Not some mnemonic or cryptography standards.