Hi Annie,That idea for the pass it on activity seems like a sweet idea for a cooperative game and including a mantra. I think, though, it does get confusing for children to be asked to "share" when it's really "taking turns". This is sing the teddy is turn-taking. Sharing is done when something can be divided- you have some and I have some, as in sharing playdoh or blocks. There may be only one red marker - then we take turns with it. The language the educator uses should reflect what the action or behavior is so it is clear to the child - taking turns implies you will get it back. That's very different from "sharing".
I hope this is helpful.
I would recommend checking out the peacable kingdom board game series, I used them with my students because it is set up as players versus game instead of player versus player so everyone works together.
These aren't necessarily for sharing but are great for self-regulation which is a key component of sharing! Great videos, 5 games in each video:5 Incredibly Fun GAMES to Teach Self-Regulation (Self-Control) | Social Emotional Learning
5 (More) Self Regulation Games and Activities | Social Emotional Learning