Not so. It was reported here by a person who went in to be tested for COVID. She went back five times and was listed as tested for COVID five times in the records when she should only have been recorded once.
You get recorded as 5 tests; one case. That accurately reflects the situation. You did - in fact - have 5 tests. You are - in fact - one case.
Cases and tests are recorded separately and for exactly the reason you’ve given here: you can have many different COVID-19 tests done, but you’re only one individual. There is a point at which one person can be recorded as multiple cases if they test positive after 3 months of no positive tests. At that point, you’re hit the re-infection lottery (about 1 per 10,000 cases last I saw).