@tjdog, You are correct. A person may have multiple positive tests, but they should be calculated as a single positive case. The number of positive tests in a state is not equal to the number of cases, as one person may be tested more than once. It would appear that the people you are talking with are confusing the terms "test" vs. "case."
See this information:
- CDC https://www.cdc.gov/coronavirus/2019-ncov/cases-updates/testing-in-us.html

There are certainly limitations to the data being collected and it varies from state to state. Rather than an over-calculation of cases, it is more likely than there are more cases than are being reported since not everyone is symptomatic and not everyone is getting tested.

From Johns Hopkins
"Since confirmed case numbers may be dependent on how much testing a state is doing, it is also important to see how many tests have occurred in each state. If people who are infected cannot get tested, they will not be counted as a confirmed case in the state’s data."

"When states report the number of COVID-19 tests performed, this should include the number of viral tests performed and the number of patients for which these tests were performed. Currently, states may not be distinguishing overall tests administered from the number of individuals who have been tested. This is an important limitation to the data that is available to track testing in the U.S., and states should work to address it."
https://coronavirus.jhu.edu/testing/testing-faq/overview#why-are-there-inconsistencies-among-testing-data-for-covid-19
Ideally, each state should be calculating tests vs. cases as this graph from Johns Hopkins shows:
https://coronavirus.jhu.edu/testing/states-comparison