Answer:

Probably yes in a sense. The nurse was correct in describing how the blinding process typically works. When you're in a study, as the patient, you don't get to know what's being given. The same is true of the caregivers, like the nurse. A proper study "blinds" the patient and the doctors/caregivers from knowing whether the experimental drug or the placebo is being given to the patient. This is done to remove the possibility that we might change our care if we knew. We avoid biasing the results this way.

In practice, the nurse feeds the patient's name into a computer which manages the blinding process, then she gets a dose of "study drug" (again, she doesn't know whether it's drug or placebo) and gives it. Her impression of that is that the pharmacy is selecting from a batch of placebo or a batch of drug when they give her the drug to administer.

A more important question is whether the same lot number from the manufacturer is being used for all patients at a site. I have read reports of different lots being potentially contaminated or producing different results coming from Pfizer, and of course with J&J, they had a manufacturer in Baltimore (IIRC?) that was shut down for poor manufacturing processes resulting in bad lots that had to be recalled from the market. Ideally, we'd want to have the same lot number for any biologic drug for continuity's sake, but is that actually how it works? Do they have large enough batches and small enough volume to do that? I don't know. I'd assume it was tracked by the study's authors.

Regarding what you're suggesting in your first post about government people getting different drugs than the rest of us. 1) They're almost certainly getting different lots due to the sheer volume of demand, 2) We have an observer bias here. Your friends might tell you about medical conditions, but would some random analyst at the DIA or NSA tell you? We'd need to see actual population-level data. I wouldn't say it's impossible after watching this administration deliberately destroy this country for the past year, but I also have no evidence whatsoever that they would or are doing it. I don't believe they are doing this and I'd want to see convincing evidence to believe such a thing were happening.