Maybe you’ll find the question–answer version easier to understand:
Qu’est-ce qu’il a de plus beau?
C’est le poil !
To formulate an affirmative sentence directly (without the question being asked) one cannot escape this phasing :
C’est … qu’il … de plus … .
Ce qu’il … de plus …, c’est … .
… est ce qu’il … de plus … .
It could also be preceded by the following complement, which could improve the readability of the sentence and does not change its meaning.
De toutes choses, c’est le poil qu’il avait de plus beau.
With the same question–answer reasoning you can find out that the following sentences are correct as well :
C’est le poil qu’il avait de beau.
C’est le poil qu’il avait de plus beau qu’elle.
And you are perfectly right to notice that the two last sentences are about one single thing that is beautiful or more beautiful for the dog than for the bitch. It is not about the dog’s most beautiful feature (characterized by a superlative) anymore.