I think there's a work around on this - I was using duplicate names for categories and, again, it's essential for me that a child of one category could have the same name as a child in the other.
I've managed to get around this by ensuring that the Category Slug for each is different. That seems to be the element that WordPress is checking for duplicates. So make your two new categories but put a spelling mistake or something in the category name of the second one. When it's made, go back and edit the Category Name to the words you want but make sure to also change the Category Slug to a corrected version with extra words in it - maybe a reference to which parent it refers to or something like that.
So in your example, you could make a child category Summer in parent category Photos, no problem. When making the second, initially create it a Sumer in Video. Then edit the category once it's made, correct the Sumer to Summer and rename the slug to something like summer-videos.
Seems to work.