He is wise among hobbits but among Middle-earth, i do not know if he has lived there long enough to be elevated to be considered "The Wise" as Tolkien has stated. Being a Frodo fan, I would say he's changed during the journey, garnering more wisdom but wouldn't say that he would be placed in the same context as the elves or wizards in terms of wisdom. However, he would definitely be the most influential of all the hobbits and he has many firsts under his belt of being the first and only hobbit to do things that others in any race could not do, which is bear the Ring.

His change of status crossing to the West is indeed another first and somewhat of an elevation of status but i'm not sure if that's considered being considered one of the "wise". He could be considered a great hobbit but in order to be considered "wise", doesn't time and experience give you more of that so therefore the aforementioned elves and wizards have that status but perhaps Frodo not yet?

This is an interesting perception. I love creating discussion with this. :)