Why doesn't "de to børn er samme gamle" work?
Because that means "The two boys are same old". If you want to use samme, you have to say "The two boys er samme alder". Gamle is old and alder is age :).
Could one say "den samme alder" in this case?
I gave that answer and it was wrong - I reported it. I think what they are using here is another idiom, because it actually says that the two children are 'equal old'. The creators of this Danish tree seemed to love throwing idioms at us.