Tap water in Germany is actually subject to even stricter controls than mineral water - that's the thing a lot of people don't know. With mineral water, only the water itself needs to be tested, but the bottling and storage are less regulated. Tap water, on the other hand, is constantly monitored from the source all the way to your home. So your friend is totally right with her basic point.
But the plastic problem is the real sticking point, and you shouldn't just brush it aside. Buying mineral water by the bottle is an environmental disaster - the whole production, transport, disposal. Even if you recycle, it's a massive waste of resources. If plain water alone seems boring to you, you could get a SodaStream or something like that, then you've got sparkling water without all the plastic waste.
On the minerals argument: yeah, mineral water has more minerals, but honestly in amounts that aren't relevant for your health. You get that much better through normal eating. But the psychological effect counts too - if mineral water just tastes better to you, then drink it. But you should be aware that you're paying for it, not because it's healthier, but because you like the taste.