Is it permissible to respond to a Muslim's prayer for Allah to provide him with a specific animal?

A Muslim asks Allah Almighty for all his needs, be they great or small. This includes asking Him for the strap of his sandal and for salt, as stated in the Hadith: "Let each of you ask his Lord for all his needs, even for the strap of his sandal."

And as Aisha, may Allah be pleased with her, said: "Ask Allah for everything, even for the shoelace, for if Allah Almighty does not facilitate it, it will not be facilitated."

So, a person asks Allah Almighty for whatever he wishes of the good things of this world and the Hereafter. This includes asking Him to provide him with an animal permissible to own.