Is there any formal understanding of how humans perceive humanity/vitality in other entities? When humans watch animations which contain human-like characters in them, we seem to imagine that the animated individuals, whether they be photorealistic or mere stick figures, are human, or at least living, in some way. If I see a stick figure get his or her heart broken, I feel for them as if I were watching a character in a movie get their heart broken. 

There is obvious nuance in that my emotions will likely be different in response to watching two situations which ceteris parabus only differ in that one has stick figure characters while the other has high-qualtiy photorealistic ones. However, there still seems to be something going on where the mind imbues entities with a perception of humanity, or at the very least of vitality.

So, my question is first and foremost whether or not there has been any research done our perception of other things as human/alive (this seems like something that would at least happen in child developmental fields), and if so, is there a particular region of the brain and/or pathway/mechanism that we know is involved with this behavior?
Yes, there's a lot of work on this! Still a ton of open questions though, and plenty of smart people working in quite disconnected paradigms.

A classic study in this space is Heider F. &amp; Simmel M. (1944). An Experimental Study of Apparent Behavior, The American Journal of Psychology, 57 (2) 243. DOI: 10.2307/1416950

But note the date! 1944! I think what you're really looking for is the community that is still citing this today.

One recent example might be people interested in Bayesian Theory of mind, eg:
Baker CL, Jara-Ettinger J, Saxe R, &amp; Tenenbaum JB (2017). Rational quantitative attribution of beliefs, desires and percepts in human mentalizing. Nature Human Behaviour

Also, 'agency' is one of the areas identified/investigated as part of core developmental knowledge by Spelke and collaborators, see for example 
Spelke, E. S., &amp; Kinzler, K. D. (2007). Core knowledge. Developmental science, 10(1), 89-96.

This answer doesn't really do justice to the breadth of this literature, there's a lot of it! But hopefully the citations and cited-by links from these will throw up the kind of thing you're looking for.