Either 'fur' or 'coat' is possible if you're talking about a Shibu Inu. 'Fur' is specific type of soft, thick, warm coat. Most animals have a coat of some kind, but not all have fur.
There are a number of grammatical problems with your sentence.
- You need to have an article before the dog breed - this individual is only one example of this breed, so it has to be preceded by 'a'.
- Unlike coat (which is countable), the noun 'fur' is uncountable, so there shouldn't be an article before it.
- The adjectives before the noun should not be connected by 'and'.
So, you could say either:
The stray dog looked like a Shiba Inu and had beautiful yellow fur.
The stray dog looked like a Shiba Inu and had a beautiful yellow coat.