Mentioned in the same breath as participation mystique, I’ve noticed is the term “sympathetic identification” which is attributed to somebody named Ernst Cassirer. No site seems to have a really detailed description of what he meant by it though, unfortunately. This page about characteristics of primitive religion mentions it in passing. This page about Druidism also mentions it only in passing.

This other page mentions Cassirer, but also attributes the term to somebody named Jauss:

Viewers assume a role within the play of action portrayed, relating to the character as a surrogate for oneself. Described by Jauss as a ’sympathetic identification,’ viewers position themselves in the place of the protagonist of the drama. A degree of solidarity results with the suffering hero figure.

And along those same lines, another site has a reference to it in relation to suffering:

As humans, we seem to be “hardwired” to compassionately respond to the image of a suffering human–something that both confidence men and public relations people recognized long before it was scientifically confirmed by the sociologists. This has been attributed to a “sympathetic identification” with the sufferer, a “like me”, and “There, but for the grace of God…” reaction.

That’s all I can find on this offhand though.