Are there any scientific studies on the effectiveness of agile (or TDD), or is this just a homeopathy situation?

Yes (and yes) on TDD. None of the ones discussed in Making Software: what really works and why we believe it were done particularly well, and the less bad ones tended to show no effect or contradictory mixed results (ie, disagreed on what went better vs what went worse).

"Agile" is far too ill-defined to actually study usefully. Specific variants or individual practices could be studied.