I think of it as a competitive process, and a bit of statistics. If a person is exposed to both delta and omicron on the same day, the omicron if going to infest much more rapidly, making the prevailing virus at the time of sampling be Omicron. It’s easier to transfer, and more rapid to replicate. While Delta may still be present in the background, it’s essentially drown out. I assume the same thing is happening on a cellular level, there are so many Omicron that the Delta can’t gain access to the binding sites, and can’t gain access to the ribosomes to replicate.
Otgerwise we’d have super infections, where the body is preoccupied with the more multiple Omicron viruses while the more virulent Delta reeks havoc on the organ systems. This would presumably test as Omicron but act like Delta, or worse, clinically. But this hasn’t happened, so I think it’s a numbers game.