Conceptually that is quite true, though the domain dependencies make a lot of the code end up looking quite different.
But the first true database (pre-relational!) was developed for SABRE, American Airlines' computerized reservation system, in the early 1960s. Before that tickets were issued manually and the physical structure of the desks and filing systems used to make reservations reflected the need!
Unfortunately I can't find the paper I read (back in the mid 80s) on the SABRE database but I remember that record size (which is still used today!) was chosen based on the rotational speed of the disk and seek latency. Certainly there was no filesystem (the concept of filesystem barely existed, though Multics developed a hierarchical filesystem (intended to be quite database-like, as it happens) around the same time. The data base directly manipulated the disk. I don't know when that changed -- perhaps in the 1970s?
Like I said I can't quickly find the paper on the topic, but here's a nontechnical discussion with some cool pictures: https://www.sabre.com/files/Sabre-History.pdf. A search for "American Airlines SABRE database history" finds some interesting articles and a couple of good Wikipedia pages.