Corki = Mario. End of discussion. Cannon.
Basically what happened was Mario had a son and his son had a son and technology got advanced.
Goombas turned into Teemo shrumz Toads turned to Yordles, Bowser became Rammus.
Yoshis were basically killed off by all of the bad peeps.
Yoshis eventually over time became a URF.
Think about it URF had thumbs, can swim can fly just like Yoshi. Corki even has a skin where he rides Urf.
His plane shoots fireballs like Mario + Yoshi and even has a long leap like Yoshi.
Corki is basically just a 10 generation down version of Mario who built a machine to get back at all the people involved in the extenction of the Yoshis 'aka Urf' He read books that his ancestors wrote and pictures of Yoshis sketched on papers. He designed his plane as much to be like Yoshi as possible.. Although Corki does not possess the fire throwing ability that both Mario and Yoshi possessed he decided to build the machine to look and even shoot fireballs just like his ancestors used to be capable of. To honor his ancestors legacy, Corki grew out his mustache.