You and she have different personalities... and different ideas about play.
Which is fine, but recognize that.
Let her develop...
She is normal.
She is age appropriate.
Both my kids do that too... still.
My kids are creative and outside the box... and they have learned how to organize and help around the house. NO, my house is not a neat freak home nor perfect... but we keep it at least neat as we can.
Next, no child, HAS TO share everything with their siblings. EACH child, should be able to keep some things just special for them. And other things can be shared. Children have to learn... that they CAN have special things that is special in their minds.. and not have to relegate everything to a generic sharing mode. THAT is not fair.
She is older... so naturally she "thinks" the play room stuff is hers. But, if you give her a chance, you will see that she understands very keenly... and will share and loves her baby brother.
(Adding this): And the thing is: for the past 3 years, the play-room & everything WAS only hers. So, in light of that, and her age... there is nothing "wrong" if she "feels like she wants that stuff to be hers...." The baby has only been around for 9 months... and in light of that... your daughter is still adjusting to everything.
Another way to look at it is: ever know a college kid/friend... who goes away to college. Meanwhile his/her parents completely change their bedroom into a study or a spare guest bedroom or an exercise room? THEN when they go back home on vacation, "their" room is NO longer "their" room... but they still "feel" like it is there room... and they resent the changes made to it... and the fact that they have to share it and have NO input into it all? - well, that is the way a mere child can feel too.... and adults. It is "theirs" and those are the feelings she has.... its okay. It is her age and she is being really great about it... if that is the only problem she has, that is not understood...
You don't have to "interpret" her behavior... she is being a 3 year old. She is totally normal and sounds like a real joy.
My friend, has 2 kids. We go to their house, and you can't even tell she has kids. Her home is totally "adult" like and not a bit of kid things around anywhere. Lo and behold.... she has 2 kids! BUT, if the kids want to play... they have to (1) ask permission first (2) tell their Mom WHICH toy they want, from the closet & only the Mom can get it for them. They are not allowed to open the closet/get the toy themselves (3) ONLY 1 toy at a time, and (4) once they finish playing with said toy, they HAVE to put it back into the closet, close the closet door... and that's it. Her kids, are like robots. Total lack of free impulses or creativity. They only go by rehearsed sort of routines. Its really kind of sad. Not real child like, nor a child friendly home. But that is her style.
(not saying this is you, but just an example).
ALL kids make piles of stuff they are playing with. It is NOT "hoarding." Be careful not to label her that way... or displace it onto her.
She is only 3. They do things like that.
For me and my 2 kids... I have a lot of cool pretty baskets around the house... that way, it makes it easy for my kids, to "help" clean up... and they can just dump everything in the baskets when they are done. Fine. If things gets left around, that's fine.
The MAIN thing... is that the child "try's their best."
Anyway, your daughter is normal.
All the best,
Susan