| This answer is opinion-based.|
Due to lack of canonical information in the franchise, there is no definite answer to this question.
This is a hard question to answer. In terms of closeness, I might suggest Tigress as they have two close moments in KFP 2 and more in Kung Fu Panda 3. Tigress is arguably Po's biggest idol in KFP 1, and certainly she was the inspiration for Po's love of Kung Fu as of Secret of the Scroll.
However, Po is shown to have a great friendship with Monkey. The two are both similar in joking personality, though this is mostly built upon in Legends of Awesomeness.
I don't think you can include family as friends for this topic, otherwise it would clearly be Mr.Ping.
Collectively, his best friends are the members of the Furious Five. He is good friends with them all and does not seem to have selected one as his "best" friend.