Even we Germans must have a prescription from a doctor to get tested; you cannot just send your blood to a University lab like you can in the case of IMD.
Yes, sometimes labs may require a doctor to order your tests. I know one person who tried to get a test at the HPI in Greece, and the HPI told him he needs to get his doctor to order it. But another person who used HPI was not told this.
I think sometimes it may depend how you come across: if you seem to know what you are talking about, medically speaking, they may serve you. But if you sound like you do not know much about medical matters, then maybe they will tell you to get a doctor's order for a test. Maybe that's how it works, I don't know.
When I got a CVB1 to 6 antibody neutralization test at the University Medical Centre of Utrecht, in the Netherlands, they did not require a doctor's order. (Unfortunately Utrecht stopped doing the neutralization test).