Skylanders side-steps this issue very well with everyone only calling you 'Skylander' and the Infinity series has characters 'locked' into a specific playset (giving you a core 'cast' for that world making their stories make sense).
Dimensions doesn't really side-step this issue and the other characters feel out of place in the core story. You don't see Ninjago helping out the gang during there level, they are just side characters in the stadium. Homer has some funny moments in his level but thats it. The Wicked Witch is the main boss of the first level; playing as her to....fight her is very odd.
Overall, its a flawed but interesting effort in the toys-to-life genre. But they have to step their game up for the next version (if there is one).
I'm sorry but having everyone completely ignore and not take notice of who you are playing or having characters locked into specific playsets, don't seem like good solutions to this problem at all, they seem like restrictions. I personally prefer Lego Dimensions decision to give the player a very structured set of story missions using the characters in the starter pack, than (to my limited understanding) Skylanders extremely devious business model of gating certain missions that seem to be part of the story.