Think of the terms as describing how the food is "packaged" or presented. "Eat-in" means it's served on plates for on-premises consumption. "Take-out" means it's put in containers so you can take it with you to another location to eat it (or have it delivered there).
Fast food restaurants are routinely designed to support a high volume of business where the food is eaten somewhere other than the establishment. In some cases, everything is packaged for take-out, but the establishment has some seating where people can still eat it there, from the take-out packaging. Other fast food restaurants will put the food on plates if you plan to eat it there.
At those restaurants, it is necessary to ask at the time the food is ordered as to which type of packaging the kitchen staff should use (wrap and bag it or put it on a plate). Eating-in may also have secondary implications, such as whether the wait staff will need to provide you with any service.
"Eat-in" and "take-out" are somewhat slang terms, and they are commonly used in different constructions and with variations in the precise terms.
Eat-in or take-out?
Do you want to eat in or take out?
Do you want that for here or to go?
Do you want to eat that here or get it to go?
Eat-in or to-go?
Regardless of the precise wording or sentence construction, it means the same thing.
Keep in mind that jobs at fast food restaurants are usually considered entry level jobs. They are not typically staffed based on a criterion of English proficiency or expertise. Even "native speaking" staff, often should have "native speaker" in quotes. "Ethnic" fast foods restaurants are often small businesses started by people for whom English is a second language. So the wording used should generally not be taken as defining or complying with any form of formal English rules or practice. Just be happy if you get what you ordered.