(It came out sometime in the last 15 years.) I think it was like a furry blue puppet and the father was obsessively talking through the doll throughout the length of the film. His wife eventually leaves him and takes their son with her in dark dramatic kind of way. It’s kind of a dark comedy I think and I’m fairly certain it came out sometime between 2005 and 2012, but I could be mistaken. I know the dad was at least a b list celebrity.
The Beaver (2011) – https://www.imdb.com/title/tt1321860/