Alsase and lorraine belongs to France or Germany?

Germany. And if you think about it France belongs to Germany too. It was part of the Frankish empire, which was created by Germanic Franks. Its name is derived from it.

That depends on how you define "belong".

The region switched countries several times.

However, the Alsässisch dialect is clearly germanic

Historically it's Rome's, and they quietly lay claim to it to this day.

People lived there for thousands of years before there was a France or Germany.  Much of Europe's changing borders are about wars and arbitrary power politics more than who belongs where or who owns what.

Historically, that question was the reason for many wars.

Neither one, actually; historically they were originally part of Belgium.