I did an internship between degrees. I was technically not a student at that point. So it's definitely possible.
Make sure you figure out (if needed) insurance, etc. That was the main complicating factor for me, I didn't have any insurance anymore for that period of time.
Does the employer have a legal obligation to hire enrolled students only? (In the United States)
Employers can hire whoever they want. People will frequently do "internships" even after graduation (or without a degree). In general, "internship" means more "shorter term duration work for early experienced employees."
Some places seem to use "internship" to underpay people otherwise doing FT work. Occasionally they are unpaid.
As stannius points out, this may be complicated if you are in the USA on a student visa (and not an American citizen).
Does the interpretation of such a requirement vary from employer to employer?
Most employers will not care about this. Their primary concern is if you say, "I am going to do degree X" that you are actually doing it.
Unless you run into an incredibly pedantic HR department, most will not care if you are doing that program in the future or currently.
You may get questions about why you deferred enrollment and the answer to that might matter. It probably will matter more than the nuances of whether you are "currently" a student or "in the future."