I guess "meanwhile" refers to time (only?), like you said "at the same time". But "by the way" refers to some events or situations you have to describe right now, I think because it contains the word "way". That's how I understand it.

By the way, there is the same issue in Russian.

By the way = кстати, между прочим
Meanwhile = тем временем, в то же время, между тем