Usually, in a private chat session, you talk about a sensitive topic which is sexuality. This would make sense not to associate your real details with sensitive information about yourself.
A woman that you will never see again
Is it worse telling a woman your real name if you see her only once in her life. Sharing your real name is useless.
She is also lying
Many women don’t tell their real first names. They often use a nickname. In fact, you never know if a woman tells you the truth online. They also often lie about their country of residence. For instance, many Russian women never tell they come from Russia. They say they come from Estonia, Latvia, Lithuania.
Protect yourself
You never know what people can do with your identity details if they are able to find it somewhere. Someone can blackmail you. You must be cautious about it. You never know what somebody can do online.
Sharing your real name will not help you benefit for a better user experience during a private chat session.
What to do?
If you are concerned about the morality of giving your real name online, here is what I have to say about it: You can tell her a fake first name at the beginning because you don’t know the woman you talk to. Even if she sounds very nice. Then, later on, if you see that you can trust her, you can give her your real first name.