The hogyoku having sentience and rejecting Aizen as its master could've gone somewhere but it didn't so Hogyoku learns the Power of Friendship AU.
Urahara's hogyoku learns about friendship via Rukia and Ichigo, Aizen's learns it through Orihime and Ichigo. About a year after Aizen's defeat, it separates from him and creates a semi-transparent body from energy and disappears into the living world.
It tracks down Ichigo and attaches itself to himself. It typically curls around his neck but as long as its close to him it's happy. Ichigo himself is just confused why the ferret-dragon-purple-noodle thing is following him.
Its body can only be seen by those it wishes to be seen by so it typically hides its "heart", the hogyoku itself, in Ichigo's shirt.
After spending time with Ichigo, it eventually learns how to speak.