The word for "mother" in Hebrew is originally אם (em). As a result of influence from Aramaic, the more used form now is אמא (ima), but אם is still used in formal speech and in constructs and expressions.
Their mother's language would be השפה של אמא I believe, but that's a different thing from "mother tongue". Usually, one would share a mother tongue with one's parents, but not necessarily. שפת אם is specific to one's mother or native tongue, it doesn't (necessarily) mean the language one's mother speaks.
What's the question? A mother's language is the language of the mother. A mother tongue is a native language. Example: my mother's languages are Yiddish and English. My mother tongue is English. (Personally I think tongue sounds weird, and instead just say native language: my native language is American English).
Is that spelled the same way as the word at the beginning of a (formal) question?