Islam encourages kindness to animals and prohibits mistreating them. The Prophet Muhammad (peace and blessings be upon him) taught that people should be merciful to all of Allah’s creation. It is reported that AnasIbnMalik (may Allah be pleased with him) had a young brother who had a nightingale. When the bird died, the Prophet (peace and blessings be upon him) noticed that the Anas‘ younger brother was very sad, so he joked with him, and said to him words that implied approval of his keeping this bird. The Prophet (peace and blessings be upon him) said to him, “O Abu `Umayr, what happened to the nightingale?”(Al-Bukhari).
Sheikh Abdul-WahhabibnNasir Al-Tariri, former professor of Shari`ah at Imam Muhammad Ibn Saud University in Saudi Arabia, states the following: It is wrong to name an animal with a name that is worthy of extra respect in our religion, like the names of the prophets or names that contain the word “Allah,” such as `Abdullah. This could be an insult to those names that are worthy of extra respect as a matter of faith, however unintentional the insult might be. We are supposed to show due reverence to the symbols of our faith.
Allah says, [And whoever respects the signs of Allah, this surely is (the outcome) of the piety of hearts] (Al-Hajj: 21:32).
also, it would be prohibited to name an animal in such a way that a specific person is clearly being belittled or insulted, since deliberately belittling people is unlawful in Islam.
Allah says, [O ye who believe! Let not some men among you laugh at others: It may be that the (latter) are better than the (former): Nor let some women laugh at others: It may be that the (latter) are better than the (former): Nor defame each other, nor call each other by (offensive) nicknames: Ill-seeming is a name connoting wickedness, (to be used of one) after he has believed: And those who do not desist are (indeed) doing wrong] (Al-Hujurat 49:11).
Aside from that, there seems to be no evidence that there is any harm in naming animals with whatever types of names that are deemed appropriate and inoffensive according to the customs of the society where people live.
The Prophet (peace and blessings be upon him) used to name his animals. For instance, he named his donkey `Ufayr. (Al-Bukhari and Muslim)