Does character selection affect the stats in any way?

From MarioWiki:

Just as in Mario Kart Wii, each character gets their own specific stat boost that accounts for kart customization, based on their weight class. The units are out of 6.[2] In general, light characters have high values ​​of Acceleration, Handling, and Off-Road, while heavy characters have high values ​​of Speed and Weight. Hover over the weight column to see the characters in the represented weight class.

Here are the different characters unlocked from the beginning and their weight classes:

Feather:
Toad, Koopa Troops

Light:
Peach, Yoshi,

Medium:
Mario, Luigi

Cruiser:
Donkey Kong

Heavy:
Bowser

Unlockable characters:

Feather:

Shy Guy, Lakitu

Light:

Daisy

Medium:

Mii

Cruiser:

Rosalina, Wiggler

Heavy:

Wario, Metal Mario, Honey Queen

Source