Language suggestion: "federal X" usually means "it belongs to the federal government". What you wanted to say it that it is subject to Bundesland ("state") control.
You are right about education being run by the individual states. However, when looked at from the outside there are still many similarities between Bavarian and Bremen.
Just like there are many similarities between shopping at German hard discounters (despite them being run by totally different companies) vs shopping at big box stores in the US (who are also run by totally different companies).
Kindergardens are pretty relaxed so. I can recommend the church run ones in Bavaria, despite the name they don't care about religion beyond some stuff around christian holidays.