As far as I care, Yoshi is the first Yoshi you meet in SMW who lives in the house on Yoshi's Island, the character from the spin-offs and Smash Bros., and just like red Kirby is Kirby, Luigi in Waluigi's clothes is Luigi, and Lucas with Claus's hair color and clothes is Lucas, it's Yoshi no matter the color scheme.
But there're more of them on the screen!!
Well, so? You can have multiple Foxes and they're all still Fox.
But you've set up Yoshi talking to other Yoshis/Pikachu talking to the one with Red's hat before!!
Yes and that doesn't make them real separate characters any more than Ganondorf with a white beard is actually Silver Neelsen or Kirby with Donkey Kong's power is actually the Bubble Monkey. If you're going to take my musings as canon, go ahead and believe that Nana and Zelda are Wario's girlfriends.
Same for Toad, the SMB2 character (despite what color limitations did to him there), the spin-off character, he who first informs Mario and Luigi of Peach's kidnapping in Super Paper Mario, the Toad who goes to Mario's house in Super Mario RPG, and I think there was one Toad who hung out with Gadd throughout Partners in Time as well. He's probably the shield Peach uses in Melee and Brawl, too.