genuinely asking, how do we actually know the delta variant is real, and is the dominant variant causing the current cases?
do people actually get tested for specific variants, or is it just a generic covid test?
It depends where you live. The UK was an early[0] leader in sequencing a lot of their cases. They hit 600,000 sequenced cases in July[1].
That said, if you just want to know what proportion of the population has delta, you don't need to sequence that many cases. The point of sequencing a large fraction of cases is to catch emerging variants, that wouldn't be likely to be caught otherwise. But to know whether Delta is 30%, 50%, or 70% of cases, you just need a good random sample, not a particularly large one, and I imagine you can do that at the labs where the PCR tests are being run.
(at one point, I recall that the UK had a test that matched on three distinct sites on the virus, and they found that one of those sites would turn up negative for a particular variant, so they were able to use that as a proxy for the spread of the variant. I don't remember if that was Alpha or Delta though, and obviously it's no replacement for sequencing)
NZ had until recently sequenced 100% of its cases, and used that sequencing information to map person to person infection linkages. This included all of the people in the mandatory 2-week quarantine at the border. So in NZ, yes, every case was tested for specific variants. Delta does seem to be real.
Some small subset of COVID tests are sent to larger labs to perform a more detailed genetic test to determine specific variants. The prevalence of different variants within that set is extrapolated to the population of the region from which it originated.
If the lab finds that 87% of the samples from Adam's College are of the Omega Mu strain, then it's assumed 87% of total cases at Adam's College to be Omega Mu variant.
In Washington state, a percent of positive tests are sent to the dept of health for variant sequencing. You can see the published results by week. Delta has outcompeted every other variant for several months. All covid is essentially delta.