The bastard names you're referring to are in replacement for family names, so children outside marriage don't bring shame on the family.

In both the book and the series it's suggested that only parts of Westeros, not essos use family names. (The wildings are natives of Westeros and don't use Family names)

In Essos most folks don't even seem to know Jorah Mormont has a family name and call him Jorah the Andal.

Note the conversation between Shae and Varys

Varys: If you let yourself believe that a foreign girl with no name could spend her life with the son of Tywin Lannister...

Shae: I have a name.

Varys: You have one name. As do I. Here only the family name matters.

The bastard names you're referring to are in replacement for family names, so children outside marriage don't bring shame on the family.

In both the book and the series it's suggested that only parts of Westeros, not essos use family names. (The wildings are natives of Westeros and don't use Family names)

In Essos most folks don't even seem to know Jorah Mormont has a family name and call him Jorah the Andal.

Note the conversation between Shae and Varys

Varys: If you let yourself believe that a foreign girl with no name could spend her life with the son of Tywin Lannister...

Shae: I have a name.

Varys: You have one name. As do I. Here only the family name matters.

Update: In direct answer to the question, What is a bastard in Essos called? By his or her given name without indication of birth in or out of wedlock.