Can I call a person who speaks the same language as mine, a language mate? Considering that's our mother tongue and we're from different countries. For example, I'm Brazilian and the other person is Portuguese and we both speak Portuguese.
There is not a common word for this in English. Your invented term of "language mates" will probably be understood, so feel free to use it.
You could also say something like:
Happy holidays to all my fellow speakers of Portuguese.