Even though Yoshi is categorised as having his franchise and in fact has his own universe in Super Smash Bros., I felt that Yoshi is still strictly a major Mario spin-off rather than his own franchise, unlike Donkey Kong and Wario. It's similar to how Peach and Captain Toad having their own games still meant that they are Mario spin-off games rather their own series.
My thought is this: Yoshi's earlier games share the spotlight with Mario, who is actually the playable character. For example in Yoshi's Safari, who is the one shooting? It's Mario! Yoshi's Cookie's single player is controlled by Mario and Yoshi for two versions (GB and SNES) but in the NES it's all Mario. And then there's Yoshi's first game, where only Mario (and Luigi) are playable while Yoshi sits in the sidelines. Even when Yoshi truly struck out on his own in Yoshi's Island, some Mario enemies still appear in his games, not to mention that it has baby versions of established Mario characters like Baby Mario and Baby Luigi. Shy Guys are the main enemies of the Yoshi games, but they still appear in Mario games despite not being regular Mario enemies.
Yoshi's main villain is even Baby Bowser, which is basically Bowser's infant self. With Bowser Jr., both of them resemble each other so Baby Bowser isn't quite looking so original since Bowser Jr. still appears in Mario games. Plus, Yoshi still has regular appearances in Mario games. Let's not forget that Smash Ultimate categorise a Super Mario World stage as a Yoshi universe level even though it's actually a Mario stage. Basically, Mario and Yoshi games still share a lot of things between them that calling Yoshi his own franchise didn't feel right.
Thank you for reading.