"Have a seat" is generally either an offer (e.g. "Please, have a seat") or direction/instruction (e.g. "You're late; have a seat"; "Take a seat" is interchangeable for either example.
"He/she had a seat" isn't ordinarily the same as using got or took - it's more typical for had to be used in a more complex phrase, like "He/she had [a seat] reserved ahead of time", or for something that is consumed, like "He/she had a drink."
- The strangeness would be even more pronounced with "They had seats [by the window]" - without additional context, this implies that at some point, they were in possession of said seats, or that they are/were already seated in those seats, but isn't technically stating the act of acquiring the seats;
- whereas "He/she took a seat"/"They took seats" is quite normal, and is completely interchangeable with "He/she got a seat"/"They got seats" (with or without specifying the location of the seats). Both of these clearly convey the act of acquiring, but had normally refers to an acquisition that has already occurred.
Additionally, "Get seats" or "Get a seat" would be most appropriate to a request/direction given by Person A to Person(s) B(+) in a situation where
seats are not assigned, i.e. 'general admission' or 'first-come-first-served';
Person B will be arriving sooner than Person A
Hence, "Quickly, go get seats while I park the car" or "Would you mind getting us some seats near the aisle?" (Save and grab are also frequently used in this context.)