The song is Adia by Sara McLachlan. The instrument I’m referring to can first be heard at 1:17 and again noticeably from 1:24 to 1:25. It’s overshadowed by the piano, guitar and drums, but it’s still noticeable. Is this perhaps an electric piano like a Fender Rhodes or Wurlitzer, put through some kind of chorus effect pedal?
Yes, exactly. It sounds like an organ that’s heavily “Leslie’d” - I.e., sound like it is being put through a Leslie rotating speaker (as it sounds like you’d know, a standard pairing for rock organ) with the rotation super high. Since it’s probably a digital effect, they just set it higher than normal parameters.
I’m being too specific, there are many ways to do it, but the point is you’re correct, there’s some tremolo-y, envelope-y effect in there.
Sounds like a guitar to me, especially at 1:16-1:18.
…or I could be wrong!! 
The credits only list an acoustic guitar on this track. Keyboards (other than Sarah’s piano) are listed, and played by Pierre Marchand.
Interesting. Could just be a random synth patch, although it sure sounds plucked to me. More investigating is required!
A friend of mine says it sounds like a lap steel guitar to him. And that was without me telling him it sounds like a guitar to me, haha. Is there a lap steel in the credits, Gatopescado?
I hardly know enough about this stuff to comment and of the hundreds of times I’ve heard that song I’ve never noticed that little warbly part (that’s what we’re talking about, right?). I watched a few live versions and in that spot, in same cases it appears to just be the lead guitar doing it, but those times it sounds different. I’m going to guess, as Wordman said, it’s either a synth, there’s always keyboards around when she’s singing or, also as Wordman said, a Leslie.
In fact, I went to pull up a video of Benmont Tench showing off his before I fully read Wordman’s post and when I came back to write this I was surprised to see Wordman used the word Leslie. That was the nameplate on Benmont’s cabinet. So much of this stuff is hacked together it seems you never know what’s actually what.
In any case, about a minute into this video you can see it.
For those not in the know, when he moves away from his keyboard to a wooden cabinet, you’ll see a rotating speaker inside of it, when the camera pulls back you can see a microphone near the bottom of the case picking up the sound.
Nope. And all the songs on the album this song is from differentiates between acoustic and electric guitars. Other songs do have lap steel and upright bass in the credits. And, of course, Last Dance has Saw.
It has a decay that an organ doesn’t have. Could be synthesizer, but my guess is that it’s a piano through a Leslie or similar effect.
Here’s music that is definitely a piano through a Leslie for comparison. Echoes - YouTube