Interest rate convergence was one of the criteria for introduction of the Euro:
https://en.wikipedia.org/wiki/Euro_convergence_criteria#Crit...
Point 5, long-term interest rates
The ECB's mandate is to worry about price-stability, not the German economy
https://www.ecb.europa.eu/mopo/intro/objective/html/index.en...
Practice may of course deviate from theory, but i don't think it did much here.
German unification was 1990, we are talking about the time around 2000. Many central banks in the world lowered interest rates after the dot-com crash & 9-11 and the economic slowdown that followed.
You might argue that this caused asset bubbles, and i would tend to agree.
If it were up to me, i would add "avoiding asset bubbles" to the ECB mandate, if you want to you could regard it as a kind of price-stability mandate for assets.