When you pick them up the first time and they pop up the message that it's a 'new item' you've never had seems to be the key. It isn't important to 'have' them after that point. You just need to get that popup for each music block in a set by picking them up individually.
I say this first-hand because yesterday I bought a house in Salatu. It had a lot of stuff left behind. One of the things left behind in the house was some large bags containing a complete set of music blocks. So I picked each block up out of the bags they were in, to let the game 'announce' the new block I had discovered. And got all the Block Trophies, for what they're worth.
I got a trophy when I found a single block, and I never save blocks, I donate them. So I think the trophy is for having had all five at some time. But it didn't work that way for OP. Weird.
It is supposed to, i do believe, be that you are holding all of whatever type of music box at one time in your inventory. However, obviously, somthing is {teehee} glitchy about this one, because different glitches have had different experiences with how they obtained them.
I for one had them all sorted into different bags in my cabinet, and could NOT get the trophies. Once i had collected them all, i had to bring the bags back into my inventory in order for the trophies to kick in.
I think the different trophies behave in different ways.
Some of the other ones have triggered for me just for having held the blocks at some point of time, but not the XS one.