No. They are not numbered on a protocol level. Ordinals is an external protocol which uses a system of accounting to attempt and externally order them.

In accounting, there are many different standards to track input and outputs to a balance sheet. I haven’t cared enough to research ordinals, but I assume they use something like first in first out (FIFO) accounting and then extrapolate that through unbroken UTXOs.

Your coinjoin can still be ordered, but it’s arbitrary and doesn’t actually affect your privacy. Depending on the accounting protocol they use there are different interpretations of how sats are ordered.