Does each team...
 

...get the same number of possessions, or maybe a difference of 1?

I don't think so under NFHS at least. Team possession ends on a try for goal and begins when possessed by a player so by that virtue I'd say no.

Unless you're counting possessions in a manner different from the rule book.

Depends on how you define "possession." If it means "occurrences of team control," then no.

But could you not make a definition that doesn't include establishing "team control".

In other words, the defense's job is to end the offense's {new definition} with the ball. Giving up an offensive rebound failed in that goal.

Well consider in the NBA, the difference between the highest and lowest paced teams (number of possessions used per game) is 9 possessions/game.

It would vary a lot in high school because most states lack the use of a shot clock.

You're getting into a realm that I purposely care very little about and that is sports statistics. I'm answering by the rulebook. :D

If we're going into the land of making up definitions for this concept, you could borrow the concept of phases of play from rugby (and a lesser extent soccer). In rugby, a new phase of play begins with each ruck or muck (similar to downs in football except there is no explicit end or beginning unless there is a scrum).

So, translating this to basketball, each instance of team control would be a phase and each phase uninterrupted by an instance of team control by the opponent would form a possession.

If you used such a definition, then at first thought, yes, each team would have equal possessions. What constructive purpose this might serve escapes me at the moment.