When they were widely used, people called cassette tapes “tapes” (common) or “cassettes” (less common). I don’t recall anyone calling a VHS videotape or VCR “a VHS”.
Similarly, I have seen people recently say “a vinyl”, which wasn’t ever the way it was said. (it would be music “on vinyl” or “a record”).
Yeah, “video” was common, but “VHS” wasn’t. Maybe kids who developed language as the format was expiring in the early-mid 90s didn’t have lots of examples and just thought the letters printed on the tape were a noun.