Is the start up sound on the DB BR 182 deliberately designed to make the do-re-me... start up sound or is this a happy coincidence?
It is possible that this was developed on purpose, but I don't know for sure. What I can do is to translate a paragraph from the German Wikipedia that refers to this topic.
Our metros in Montreal do the same thing. I think it's the power itself. I know this doesn't have anything to do with the German locomotive you're talking about, but I'm certain it's the exact same thing happening (check the 0:14 mark). "The Dou-Dou-Dou sound initially came from the power converters that protects the installations from an overcharge during the acceleration phase. As it became to be associated to the departure of the trains by Montrealers, it was selected as the new sound announcing the doors closure."