In a shoot that is an assigned-type project, I am being paid to create pictures for the client's website. The models are his own/his friend's animals.
To my understanding, in the case of a human model, if he/she is not associated with the photographer or the client, we would require their model release.
Does the breeder need to grant me a model release if they are his own animals? (he would like exclusive digital/web usage. However he is fine with my making prints.)
Not really an answer, but you forgot to mention which jurisdiction is involved, rules differ between countries.
But in any case, if a release is needed, it is needed by the publisher of the images, not the photographer. (of course, they can be the same person fulfiling both "roles"). If the breeder is the one publishing the images of his own animals, he has no need to sign (or get) any release (of course, he'll need a license from the photographer for the copyright, but that's a separate issue).
But to be really sure, you'll have to consult a lawyer in your region who is familiar with those issues (not all are).
A model release is a (partial) protection against libel and animals cannot be libelled so in that respect one is not required.
That said; in some regions a property release is necessary/useful for commercial publication, so some kind of "animal model release" may be a good idea. It will give the animal owners a warm feeling and ensures that everyone involved has a clear position - this is especially useful in competitive breeding circles as relatively minor grievances can blow up into major issues.
Whilst it is the client's dime, I would argue to ensure in your terms/licence that you can use the images in your online portfolio - but that's purely down to how much money they want to give you.
Based on your related question about licensing, I'd say a separate model/property release would be unnecessary. Such details can easily be included in the contract between you as the photographer and the owner of the animals as your client.