Was it a requirement at Hogwarts that a teacher could only head the very same house which was their house when they were a student?
As we know from this answer, Minerva McGonagall was a Gryffindor herself and Filius Flitwick was a Ravenclaw himself. And of course we all know that Severus Snape was a Slytherin.
So this raises the question, was this a prerequisite for the position, or just a preference or coincidence?
J. K. Rowling has a relevant quote in the FAQ on her old website:
If a teacher is head of a house, can we assume that they were sorted into those houses when they were students at Hogwarts? Is that also true for the house ghosts? So was Snape a Slytherin?
Yes, if the teacher is Head of House you can indeed assume that they were pupils within that house. So Snape was very definitely a Slytherin and yes, the same is true of the house ghosts.
This doesn't completely answer the question, because it might only apply to the heads of house we meet in the book.
See also the relevant question What's the breakdown of Hogwart's Professor in Houses?
There is no evidence to the contrary.
The 5 heads of house we know are
All 5 of these heads were also in those houses in their time at Hogwarts initially. Whether that's coincidence, because they know the house, or because they represent the qualities of that house is not said.
Curiously, the wiki says this, emphasis mine:
Severus Snape was made Head of Slytherin at the remarkably young age of twenty-one, when he only just returned Hogwarts. It may be that he was the only teacher who was in Slytherin, following the retirement of Horace Slughorn in 1981. When Slughorn returned in 1996, Snape jealously guarded the prerogatives of the Headship, suggesting the teacher may not have to retire to lose the role.
However, as we all know, the wiki isn't fantastic and I have no idea where they get that info from.
If they were leading that House they would also presumably be a good fit for it personality-wise, so presumably the heads of the houses would be people who had either been in that house themselves or could have been sorted into the House. There was a concerted effort to maintain a particular kind of culture in a House (which was part of the point of the Sorting Hat in the first place), meaning that both the head of that House and the students who were selected to be members of it should be good cultural fits for it. In fact, that was part of the point of the House system to begin with.
The system didn't end up working perfectly, of course, but by and large people did exemplify the values associated with their house. Notable exceptions included
Professor Gilderoy Lockhart was in Ravenclaw in spite of his notorious ineptitude, and Peter Pettigrew was in Gryffindor in spite of his cowardice and treachery.
There are, of course, some people who would've fit in multiple Houses. For example, Hermione could've been in Ravenclaw, Neville could've been in Hufflepuff, and the Sorting Hat wanted to put Harry in Slytherin initially. (The main reason that Harry wasn't in Slytherin is that he begged the Sorting Hat to put him in a different House instead and the hat took his choice into consideration).
Snape himself evidently could've been in Gryffindor, at least by the time that the books took place. Dumbledore said that Snape was courageous and that sometimes he thinks that they sort too early, which implies that he was a good fit for Slytherin at the time that he was sorted but became a good fit for Gryffindor later due to his actions in being a double agent for Dumbledore. Also, Harry later referred to Snape as the bravest person he knew.
It's not clear from the books if those individuals could have been eligible to be Heads of the other houses.
TL;DR Odds are the Head of House would've been from that House whether it was formally required or not because both the students and the Head of House were expected to be good cultural fits for the House.
