I don't think anyone will be trying to collate results with names of people who ordered tests.
Maybe not, but they need to know if anyone starts ordering tests and not returning them. That's exactly why you have to have a code to order the tests in England. So they can trace you.
Alegrias1 if scientists and/or statisticians draw any conclusions from the test results they get, without allowing for the fact - which they must know - that thousands or even millions of people are not reporting, then they are remarkably stupid.
One might say the same about the people who don't think they need to report negative results even though every piece of accompanying literature tells them they have to. The metric is, how many of the tests being returned are positive. Its not rocket science.