Yes, the max level is 99. The highest you can get a stat at is 1004 if you use a Teeheespresso on that stat when it's at 998. Unfortunately, it resets back to 999 when you equip or unequip something. I have 999 in every stat for both brothers, and I have both brothers at level 99. When I was getting to level 99, I trained on most of the enemies in Joke's End. They give a lot of experience points, and the battles are usually fast if you have a lot of Attack.
I'll tell you my strategy for maxing out your stats; I'm sure it will help:
For the max stats, I put on the spike badge on both Mario and Luigi and the Game Boy Horror SP on one of the brothers. I use Teeheespressos because they give +6 in a random stat; more than any other drink. I go east of Castle Town to where the Lakipeas are. I stand slightly to the left or right of the Lakipea and let it throw a Sharpea down so that it doesn't fall directly onto me. Then, I jump on the Sharpea when it charges at me, and I go into a battle and kill all the enemies in one hit with a First Strike. Since I have the Game Boy Horror SP equipped, I always get around 2 Woo Beans each battle. Repeat until you have 99 Woo Beans. For Hee Beans, play the Super Border Jump Minigame in Stardust Fields and choose to play Super Border Jump Sunshine. When playing, make sure that both brothers get hit twice to save time at the end. Repeat until you have 99 Hee Beans. Now, go to the Starbeans Cafe and make 9 Teeheespressos. Repeat until you have 99 Teeheespressos. There are 2 things you can do right now: save and use the Teeheespresso to get the desired stat increase for the desired brother and reset the game when you don't get the desired stat, or just randomly use the Teeheespressos on the brother whose stats you want to be increased.
This takes a very long time, so unless you're devoted to doing it, I don't recommend doing it. Good luck!