Congress has passed some laws that enable POTUS (via is branch) to restrict access to certain information. For example 50 USC 3126, defines classified information to be

information or material designated and clearly marked or clearly represented, pursuant to the provisions of a statute or Executive order (or a regulation or order issued pursuant to a statute or Executive order), as requiring a specific degree of protection against unauthorized disclosure for reasons of national security.

Or, 50 USC 3164 defines it as

means any information that has been determined pursuant to Executive Order No. 12356 of April 2, 1982, or successor orders, or the Atomic Energy Act of 1954 [42 U.S.C. 2011 et seq.], to require protection against unauthorized disclosure and that is so designated

Under existing law, material is classified with a certain statutory authorization (in these instances under Ch. 44 of Title 50, "National Security"), and Congress has given POTUS the sole power to add and subtract documents from the list of "classified" materials. See especially Subchapter VI.

Of course, Congress cannot implement these laws themselves, that is the prerogative of the Executive branch (perhaps that is the feature of separation of powers that causes problems for understanding "classification"). In theory, Congress could pass a new law that imposes a special "classification" on a certain kind of document and furthermore limits the president's discretion to impose or remove said classification, they just have not done so yet.

The real limit on presidential declassification power is the Administrative Procedure Act. POTUS cannot, himself, actually do much except as commander-in-chief, besides order other people to do things. A previous president issued an order that describes what should be done by departments, w.r.t. classified material, but each department then has to devise "rules" which, depending on the rule, have to be put out for public comment. As long as there exists rules (in the Federal Register) that require something more than presidential say-so for classification or declassification, then POTUS cannot unilaterally (de)classify, he also has to follow the rules. Which he can order changed, but there is a procedure that has to be followed.