My teacher once told me when referring to body parts you do not have to say "my" like you do in english it is implied. Я чищу зубы - i am brushing my teeth. Not я чищу свои зубы. Is this true? i guess my question is when do you need to say свой in reference to a body part?
Язык is tongue, even when talking about languages it's technically tongue, think of the English expression about having a "mother tongue" and it makes more sense how these concepts are also related in Russian.
But looking away from the "parts of body" lesson (there are also sentences from other skills in the dative plural skill), could it actually mean language (I know it might not make much sense but yeah)? xD