And I would say no as it is, merely, a leftover structure from an exposition.

It was designed, with classical Greek architecture in mind, but was never intended to be a monument, any more than a local theater building built in Classic Greek style would be one.

Or a shopping mall entrance, or a Casino, or a mansion.

Yes, it is. A monument is something that was built to commemorate an important person or event, and the Palace of Fine Arts was built to commemorate the Panama-Pacific International Exposition, which was a World’s Fair held in San Francisco in 1915.

Bottom line: yes.