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.
Actually, you can use these wallet software to securely maintain and access your cryptocurrencies, NFTs, and other blockchain assets.
Your private key or seed phrase is what makes up your wallet.
I’m not sure what you really trying to prove here about the definition of wallet. All that you mention is technically functioning as a wallet since you can access your funds and manage it once you import your private key or seed phrases. Also you generate your seed phrase on one of this wallet and you can just use it on different wallet since they are just using same path to connect on blockchain.
Private key and seed phrase is by definition your keys to your wallet which holds asset on the blockchain. You can’t access your crypto assets on the blockchain without importing keys to the wallet. Having the ability to import your keys on different wallet software doesn’t mean that they are not wallet.