I think the idea is that the degree belongs to the person who earned it, therefore it is a bachelor's degree, etc. because the person who currently owns it is a bachelor. The rifle would be treated similarly if the person who acquired it and now owns it is a sniper. If it was acquired for a collection or some other purpose, perhaps sniper rifle would be more appropriate. I would be quicker to criticize the terminology of martial arts except that I personally believe that either terminology would be correct.
I think you're taking the "possessiveness" of the possessive form too narrowly. When you earn a bachelor's or master's degree you are not taking it away from anyone, it's yours. You become the bachelor or master who holds it, so it's your master's degree, the degree of a master. A sniper's rifle is one designed for a sniper. Not necessarily a particular sniper, any old one will do. If a shirt is designed for a woman we don't call it a woman shirt, it's a woman's shirt, whether or not a woman owns it.
The conventions of martial arts may very well be different from academic usages, so if you receive a master black belt it doesn't mean that it's wrong to receive a master's degree. Language and traditions vary.
Edit: Hi, Dick, you nosed me out!
Interesting question. If we're thinking of the genitive as denoting possession, the ending is on the wrong word. We say John has a doctor's degree, but in fact the degree belongs to John. It is John's degree. John didn't borrow it from some unnamed doctor who really owns it!
The phrase must come from a loose adjectival sense of the genitive, not a possessive sense. John has a degree typical OF a doctor or a degree expected OF a doctor.
I would contend that both "Bachelor's degree" and "Bachelor degree" are incorrect.
If the college issues you a certificate declaring you a Bachelor of Science or a Bachelor
of Arts, then what you have is a Baccalaureate degree.