Can’t help but notice the two ways we use the word “play”.
Regarding the etymology of “play”, the connection comes from the concept of movement (“plegian” is a Greek root for “to exercise, move briskly”). Plays were performances in which actors moved around, and games are experiences in which players can express their agency. The connection, here, then, might be that of movement and agency. (Historians, please school my ass if this is wrong)
I think it’s also worth noting that we talk about “play” sometimes as the act of parody but without simulation. As in, “playing House” or “playing Cops and Robbers” or “play-fighting”. In these things, we are doing things which represent and mimic their subject, but without the same consequences. We are pretending; we are acting. Games also abstract actions and turn them into representations of their real life counterparts. I mean, watching drama is basically watching a lot of people pretend on stage. But within those dramas, frequently there is the express purpose of attempting to more closely represent reality, which is something we also see in games. Art isn’t only mimesis, by any means, but here’s my point: both theater and games put their respective players in a space that has a framework distinct from reality’s normal consequences. (See: the magic circle.)
I want to mention improv theater, also. I know improv is kind of a cliche about college students and failing actors, but hear me out. Improv is essentially structured around actor’s agency, and often invites the audience to exert their agency to influence the performance (e.g. audience suggestions). Often improv is broken down to into “games” which structure those actions. Watching improv is a lot like watching people play a game; you are quite literally watching a “play”!