This really comes down to someone chose a name and they choose “Hudson Bay.”
A bay is a kind of sea that is mostly surrounded by continuous land. Being that it’s rather large area, it is perhaps a judgement call whether it’s a sea or bay. After all, the Mediterranean Sea is almost completely surrounded by land. An alternative name is Gulf, such as Gulf of Mexico. (Although I would argue the Gulf of Mexico is just part of the Atlantic Ocean and not worthy of being called a gulf. It is just too big. And my opinion highlights that difference is somewhat subjective.)
So I guess the answer comes down to someone chose to give it that name. (Hudson Bay is a proper noun. A bay is a type of geographical feature. Hudson Bay is a sea, bay, and gulf.)
The namers could have correctly called it a sea or gulf. That makes for an interesting follow-up historical question, “Why did they choose that name?”
I read something that suggests a bay has calmer weather than a sea, and this is a meaningful difference when you are sailing.
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