How does a spinning object know it is spinning?
Let's step back. How does an object spin? First imagine a rod, if you stretch (strain) the rod to be longer than its natural rest length then like a spring there is a force (stress) on the parts trying to compress it.
An object spins when it has some velocity in one direction and yet it the orthogonal direction it is too long (strained) so it has a stress in the orthogonal direction. You could imagine a spring with masses on the two ends. At rest it has a particular length.
When it spins it is longer and the two masses have a velocity orthogonal to the spring. It's literally longer and the parts are literally moving with respect to each other.
There is no way at all in which it is any way like the stationary spring. Just because it didn't stretch much doesn't mean it isn't stretched. A spinning object bulges at its equator, that's how it spins.
Now as for how you know. You could look at the parts and notice they are strained by measuring their separations between each other and considering the materials they are made of and how far apart their natural separations are you see that they are too far apart (that's measuring the strain). You can also measure the stress. You could also use the comoving coordinates of the parts as a reference frame and check to see if Newton's laws hold without fictional inertial forces (they won't). You could take something that moves through a vacuum at a steady speed such as light and send it around one way and then send it around another way and see if they get around in the same amount of time (they won't).
A rotating object and a not rotating object are different and there are thousands of ways to tell the difference. It's a bit absurd to even imagine they are similar in any way. Spin a spring and literally watch it get longer. What's confusing about that in the slightest?