Sounds like boss is a complete plonker
you need to arrange a meeting with him
ask him
”what, is he wanting to achieve, in terms of a specific defined objective, for this team building exercise?”
” What specific problem is he trying to solve?”
“How will he measure the “nicest” of relative actions you each make- what is the criteria and who will assess this?”
”tell him to give you examples of things you think would qualify ?”
and if the examples are things to spend money on or your own time, say clearly “ no I will not be doing that . It is not part of my job contract to spend money or my own time on team building intiatives. Is it part of yours ? “ ( it won’t be🤦♀️)
Then say to him that in your opinion this is not an effective way to build cross function teams. It will merely lead to resentment within his own team, if the other team don’t reciprocate (eg why should you do all the running) and feels a lot t like just sucking up to them. That is not going do good working relationships, it will more likely result in the other team believing thy call all the shots.
then tell him that strengthen relationships is a great idea, and wouldn’t it be far better to use proven effective ways to do team building. These things don’t need to cost money, examples would be (maybe think of few of your own so he can see you are trying to meet his objective )
sharing your lunch time in small groups to “get to know you”. Lunch and learns form team members about what their role is or specific tasks/systems they use, Setting up some cross function teams that can compete for silly quizzes , identifying who the baby was game, who has the silliest pet photo, all the way to the company paying for a drink together after work one day. it should also include lots of mutual recognition as specific people or teams reach their objectives along the course of your working togther - even just gathering folks round and him saying thank you can be mighty powerful . That does need organising, can all be done at pretty much zero cost. But he needs to a sign specific tasks to individual team members in both teams to do that.
And is exactly why he doesn’t want to do that proven team building, cos he’s a lazy arse , or just stupid, and wants you all to magically make it happen .