TL;DR: No. Pure Coincidence
All measurements are derivative straight of the turning mechanic's 10 mm diameter.
Also, everyone who has ever produced Bandsalat discovers this. It was an every day issue back then an I would know no school kid not going that way.
Basic Design Considerations

The 'hub' is not designed for a 6 sided transport pin, but for a round one pin with two 'spokes'. The hub being 7.5 mm and the spokes 10mm diameter.
The cassette side could have been made the same way(*3), but that would have resulted in a quite inconvenient handling if they weren't aligned - which they are never. To solve this the cassette got not one but three positions the transport spokes could fit when it got mounted, which equals to 6 'slots' he cassette wheel needs (3 times 2 slots).
For more we may need some guesstimate math:
With the given diameter of 10 mm this comes down to roughly 5 mm per section at the outer rim. Making the spokes of those slots about 1-2 mm in length the inner diameter comes down to ca. 24mm or 4 mm per section. With ca. 1mm wide spokes on the cassette side and 2mm wide on the drive side this comes to a 2 in 3 (66%) chance that a cassette will fit right out without any alignment issue and a slack of at maximum 15 degree (1/24th). For the remaining 33% the cassette wheel had to move a maximum of 7.5 degrees or half a millimeter.
A calculation laying the foundation for a successful hassle free consumer product ... that is ignoring all the issues bad manufacturing of cheap recorders and even cheaper cassettes brought.
Why Do Pen(cils) Then Fit That Nicely?
The fact that a pen can make a good tool to turning the tape is pure coincidental. Any hexagonal shape with diameter between 7 and 10 mm, or roughly 1/3rd of an inch, will do the trick. And that's simply what most pens feature - as it's what the human hand holds best when it's about fine control with one's finger tips. No matter if an paint brush, pencil or a bruin.
Likewise the hexagonal form of pencils is due to this being the least complex, non square form to be manufactured from a board(*3). A development dating back to the late 19th century.
Plastic pens, like Brio or BIC, in turn took that shape to sell a product that fits the expectation formed at users by pencils.
*1 - Engineers love nice strait decimal values and being developed in Europe its of course metric :))
*2 - Classic magnet tape reels had such a 3 prong design - but also no restrictions by being hard connected to another reel or in a case.
*3 - Square pencils are unpleasant to handle.