This is a Major debating point
A lot has been said theoretically about this. You have re-discovered an long running argument in the philosophy of physics.
It is often called the "bucket argument" as one of the most famous examples is a bucket of water that is spinning, and so the water level is not flat (bulges up the sides centrifugally) just like your example with the earth's seas bulging at the equator.
It is a longstanding problem amongst philosophers of physics. One of them (Ernst Mach) was especially concerned about this and his concerns fed into the development of general relativity. Mach believed that all motion must be relative, so that the centrifugal force driving the water to bulge outwards at the equator is actually caused by "the fixed stars" (somehow), so that in an empty universe with just the Earth alone you wouldn't know if it was spinning or not and the water would not bulge.
However, General Relativity failed to fix this problem. With GR in an empty universe with just a bucket of water (or the Earth), the bucket/Earth still knows if it is spinning or not.
Their are two camps on what this means:
(1) - Our best theory of physics tells us that rotation is absolute not relative. So it probably is. Stuff "just knows" when it is spinning, this needs no special explanation.
(2) - This is a really big problem, and shows that General Relativity isn't entirely right. One day we will find a better theory that fixes it so that things can only tell when they are spinning through interaction with other things.
Their are people in both camps. We may never know which group is right.