What is the Chinese meaning of giving their names to ...?

It depends on the specific sentence, it's hard to say, probably telling xx their names.

On some occasions, such as ordering something, but it is out of stock, the salesperson will ask you to leave your address, name and phone number for easy contact. They will say, please give me your name, address and phone number, which means the same as giving me your name, address and phone number.

There are similar ones.

Name sb. xx

Give it to sb. The name of so-and-so

They named their first child Roland.