My friend’s dad thought he could send me to ask my dad for a square drill bit when I was like 10 but my dad had me helping him build an airplane in the garage as young as possible. So I told him
Except it's all at once, rather than you having to drill the hole and then chisel it out. The chisel part of the bit pushes the wood towards the drill part of the bit as you drill, and then the drill part of the bit removes that wood, same as the wood the bit is actually drilling through
They do indeed really cut square holes. They're called mortising bits. Like the other commentor stated, they're basically a combination of a chisel and a drill. The drill does most of the work in waste removal while the square blades give you 90° corners.
One could also potentially consider a rotary broach to be a "square drill" (supposing that it is a square and not a hex or other shape).
Due to how my brain works (ADHD), I specifically have trouble with being concise. So, despite what you may think, I find your accurate and concise explanation to be excellent.