My dad’s really old camera took square pictures on 35mm film. I have negatives and some slides from the 1940’s and 1950’s.
IIRC old Brownies and some pro cameras used 120 film which was 2.25" square. (Although there were some that took rectangular pictures, 2.25x3.5 IIRC. Or was that 620?) The film came on metal or plastic spools with a paper backing so the image numbers (printed on the back of the paper) could be seen through a little window on the back to assure you had wound the film forward just right. 120 was also popular for those twin-lens cameras, and the large negative made it easier to get printable news photos etc.
3.5" square prints are the logical version of 3.5x5 rectangular prints when printing from square negatives. Remember all this material - especially color - was expensive back then so smaller was cheaper. I think I have some of those prints from the 60’s.