Fascination with the eagle.

[Chronos]

Well three reasons could be
1. Firstly it flies , we have always been fascinated with flying . If you look at many languages you will find sentences like soaring to new heights etc .
2. It also has a certain bit of majesty to it combined with it being the top predator , thus a combination of graceful yet brutal.
3. Early mythologies perhaps? I'm guessing early humans would be fascinated with sky and what would be above it , birds being able to fly would become part of the mythology I guess.

But an important point to note would be that almost all of these countries you mentioned have some relation to Rome.

For ex The title of Czar comes from Ceasar ,The Russian territories were followers of the eastern orthodox church and became christian due to the eastern roman empire hence the got the eagle from them. The Germans again have a relation with Rome(Holy roman empire etc) so in my opinion it was Rome's adoption of the eagle that led to it being a symbol of the majority of the nations.
Just my 2 cents

Why aren't you a think tank yet?

A really intelligent answer.