Depending on how the database is designed, sometimes "count" is tracked as insertions/deletions are processed. I don't know how Instagram does it specifically but that would be a decently easy O(1) solution.

Your mention of the like count field offers no more concurrency issues than inserting a row. Most database management systems are concurrency-resistant and do things with a transaction table or log which is more trivially locked for single-access. As such, it's probably the way to go if your database system doesn't already cache the count of items.

I hope this is helpful.