Every bitcoin output is locked to a condition, most often a signature from a specific key. The private key is the secret that produces that signature. A seed phrase is a human-readable backup from which all of a wallet's keys are derived.
Companies that hold bitcoin for customers say so in these terms. Block's quarterly report states that it "holds the cryptographic key information and maintains the internal recordkeeping of the bitcoin held for other parties," and that the customer bears the risk of loss from fraud or theft unless Block was grossly negligent. That sentence is the difference between an account and a key.
The paper's rule of thumb: an asset you cannot sign for is a promise from whoever can.