Word for people who assume the emotions of the people they're with

Emotional sponge is more colloquial than empath

For lack of a better source:

The Empath: Emotional Sponge Empaths are highly sensitive, loving, and supportive. They are finely tuned instruments when it comes to emotions and tend to feel everything, sometimes to an extreme. https://www.psychologytoday.com/blog/emotional-freedom/201102/what-is-your-emotional-type

I don't know if there's a noun form, but the adjective empathetic describes a person like this. From Oxford Living Dictionaries:

Showing an ability to understand and share the feelings of another.

So your sentences can be rewritten slightly to use this.

Even though I was aware of the tension in the room, I stayed calm. Since Aaron is empathetic, he couldn't help but get upset.

I could tell Katy is empathetic because her mood lit up when we joined the celebration.

I would call that person an empath:

empath n
One who is highly sensitive to the feelings of others and has a high capacity for empathy: "He was also something of an empath, intuitively alert, it would seem, to what was going on behind those faces" (Roberta Smith).
_{American Heritage Dictionary, TFD Online}

This feels like a rather recent coinage, but it follows the pattern laid out by words like sociopath, etc.

I'm signing up just to chime in because it's been bothering me for a while and I think I've found a very related concept called "emotional contagion" that describes a lot of what I think we're getting at, but not so much the willful action of using it functionally. https://en.wikipedia.org/wiki/Emotional_contagion

The psychological concept of "containment" keeps ringing in my head as well, and I think that gets at more of the functional aspect of what you're describing. Containing, holding an emotion for others, and then expressing it almost on behalf of them. Embodying/taking on the general emotion of a group and reflecting it in a slightly more exaggerated way has the effect of lowering the overall anxiety in the room, like the act of voicing it makes others feel that they are not unheard or alone. Certain personality types are prone to this, I would avoid the term "empath" entirely to avoid the whole illusion of personality types. It's a defense, it's an impulse to quell anxiety that has a benefit to others and you do it because of something weird in your history.