Yes, the thumb is a finger - that's why we say we have five fingers on each hand. It's certainly confusing that we refer to the index finger as the first finger; but nevertheless the thumb is a finger.
The thumbs proper name is the pollex and is not treated as another finger medically becausen it's bone structure is slightly different to the other fingers. We have four fingers and a thumb but we have five digits.
Sorry to be insistent on this, but if you look up "finger" in a dictionary, you will find it defined as one of the five digits on the hand. The definition may also add something like "especially any digit other than the thumb"; but nevertheless it is correct to say that the thumb is a finger. (I trust everybody agrees that we have five toes on our feet.)