No - this doesn't sound natural at all. I work in medical information and can tell you that there is no such term. Although terms like "man-hours" exist, "person-times" is not in use.
If the main focus of your report was the number of tests processed, then that number (using your examples) is 203,000:
The hospital processed 203,000 tests.
If the main focus of your report was the individual patients, then that number would be 2030:
The hospital tested 2030 patients.
If you want to make it absolutely clear that those patients have only been counted once, you might say "unique patients".
To explain that each patient has been tested 100 times you could say:
The hospital carried out 203,000 tests on 2030 unique patients.
Looking at the second example you gave about the class, I don't like that at all. "Class" is being used as a collective term, so saying "the class has done the test 60 times" means they have done it together, which is completely at odds with the fact that some individuals in the class have done it more than others.
I think that "class" should be used as a noun for the lesson, and instead say:
The test has been done in the class 60 times; some have done it once, others twice.