How can the 2nd local player rank up their character playing online or offline?

They can't.

Please rephrase

I think you might be mistaken. Any player, other than first player, cannot level up their characters for minions/costumes/etc. You (or the person playing first player) are the only one who can level up your character (win or lose). You might have seen someone playing online or inviting a friend, which made it seem like there was another player playing locally, but it was probably online. In that case, the second player (or any other player) can probably level up or have a level higher than one. But, I am pretty sure that any other player other than the first cannot level up and even if the did, it won't be saved. I know it is a pain, but leveling up doesn't mean anything other than how much time you devoted to a certain character or at least to unlock costumes/taunts/etc. But, once you unlock it once, any player can use it (at least the costumes, not sure about how to go about using everything else). Hope that helps!

The second player isn't able to level up. If you see a "blablabla(1)" with a rank higher than 1, than it's a glitch. It happened to me just a little while ago.

The rank of that person doesn't go up though, it just shows the rank that player 1 has with that character.

I completely understand where your coming from ThatGuyWhoWon. Me and my little cousin saw the same thing yesterday. Two accounts on the same PS3 and both had a character rank. And me and my cousin is quite sure we saw the (1) at the end of one of their names. A rank 999 Sly and rank 999 PaRappa. So I believe you, but these people cannot possibly believe because they never seen anything like it before. So you can't blame them.