Mine is similar. Also 9 years old, but has always been very sentimental. The reality is that we just cannot keep everything, and it's not great to let things build up in our home when others could be using them. I usually just get rid of things without even mentioning it (like "oh, this is your new car seat, seeing as you've outgrown your other one", and the other one just goes out without any discussion), or if I know it's something she may be funny about (baby toys etc), I put them in the loft for a while, then once they're out of mind, they go to the charity shop without her seeing.
She's actually better at getting rid of things if she can be the one to pass them on to someone we know with a younger child - like if I say to her "you could see if Emmy wants that little chair, as we don't have space for it now, and she's just the right size for it", so she has control over the "getting rid".
It's tricky, and I don't really understand it as I'm a very good declutter-er, and I hate having things lying around that are in excess of what we need, but she can't realistically keep everything, so one way or another she needs to learn a way to let them go. All power to you though - it's not fun for parents! But as PP says, we are the parents, and the stuff has to go. Best if the habits can be broken before they become hoarding adults...