Assalamu alaykum. Is that type of speech allowed, "Allah has saved that person through the doctor"?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

There is nothing wrong in the expression you mentioned as the speaker attributed the salvation of the sick person to Allah through the doctor, who was a means for the cure.

Allah knows best.