Answer:
A toy that requires some string and a breezy day
When we stand up it lies flat. When we lie back it stands up.
Small and bearded, both in the real world and in stories.
I am something that floats. But after a period of time I fall.
It looks like it's grizzly but it's actually soft and cuddly. It may be big or small, all stuffed up and ready to be snuggle.
Those with eyes bigger than their stomach will definitely leave the restaurant with one of them. It is named after a house pet as well.