In Italian, it's Colombo. In his writings he called himself Colom. In his (probable) native tongue of Ligurian, it's Corombo. In Spanish, his employer's language, it's Colón. In English, we Latinize it to Columbus. Why all the names, especially a weirdly Latinized one in English?
At the time of the discovery, Latin was the language used when writing books and creating maps. For that reason, a large number of famous people of the time are known by their Latin name. If you look at books of the time, the name in their own language is likely not even present in the book at all, so for someone to call them by their name in their own language is a guess. As you mention, whether he wants to be known as Colombo, Colom, Colon, Corombo, or something else, simply wouldn't be known to someone in the 15th or 16th century. So they're definitely going to be using the Latin last name as a result. As for the first name, though, the Latin is not used. This, too, follows a convention in which there were established first names in different languages. While spelling wasn't standardized at that point in time, it was typical to anglicize first names, so it would be common to use the English version, rather than wanting to adhere to a Latin that would sound artificial for a first name.
A related question: why do European cities have names in many languages, but cities outside Europe are more often referred by their original name (sometimes transliterated)?