Yes, "to boast" can also mean "to have" when it is something that you would be proud to have.
The hotel boasts the fastest Internet in the country.
You would NOT say:
The hotel boasts very bad service.
I that case you would say:
The hotel has very bad service.
I think we usually use it for things like hotels and restaurants. I don't think we use it for people. At least, I can't think of an example.