I'm pretty sure that this is wrong, but I just had to guess
KINDERGARTEN?
I feel most at home in Hannover
Kindergarten was invented in Germany.
But I love travelling to Dover
It quickly spread to England afterwards.
Two years long we've seen another
This could be Junior Kindergarten and Senior Kindergarten.
Sometimes along with your mother
Parent-Teacher Interviews!
Peace was made many times
Reconciling with your teachers and other slightly larger small children...
Over my people's insignificant crimes
After all those fights and my Goodness stop waking me up during naptime and teacher he stole my toys/crayons/lunch!
Hint:
When you were young you came to me
You go to kindergarten when you're young
To play, to sing, to laugh, to be
You do all of these things at kindergarten.