When making non-music recordings, sometimes I listen to a three-second snippet repeatedly and after a few listens, I cannot help but hear a rhythm or a certain musicality. What's this phenomenon called and who studies it?
This sounds similar-to (but more general than) the so-called Speech-to-Song effect, a musical illusion discovered and described by musical psychologist Dr. Diana Deutsch, whereby a repeated phrase of speech comes to sound like music. I think the effect you're discussing is a more general effect, since it involves any repeated sound, and does not necessarily take pitch into account (since you're asking specifically about rhythms). Dr. Deutsch describes this illusion (and several others) in an interview on the Composer Quest podcast here (this particular illusion is described starting around 9:45 or so).
In the case of rhythms specifically, I'd say this is actually what is called Beat Induction -- which is a form of entrainment:
Beat induction is the process in which a regular isochronous pulse is activated while one listens to music... Beat induction can be seen as a fundamental cognitive skill that allows for music. We can hear a pulse in a rhythmic pattern while it might not even be explicitly in there: The pulse is being induced (hence the name) while listening—like a perspective can be induced by looking at an arrangement of objects in a picture.
As for who studies these kinds of effects? Musicologists and perceptual/cognitive psychologists. It seems to be an open question in psychology whether beat induction is an innate biological ability, or a learned, cultural feat. According to this article in Psychology Today, recent evidence (comparing brain scans of adults and infants while listening to various types of music) supports the idea that it is innate.
A rhythm is essentially a recurring relationhip between the time intervals at which noticable 'events' in a sound wave are occurring. These 'events' can be of many types, though - could be a drum beat, or the start of a note being played - or it could be a sudden change in timbre of a sound, or even a sound stopping.
Think of what happens when a CD player can't read a CD. It starts playing the sound in its buffer round and round - typically a half-second snatch or so. Let's imagine this snatch begins just after someone has started saying the word 'so'. First, there will be a noise from the sibiliant 'S' sound, then the vocal cords will start to vibrate, so there will be a pitched sound... then it will go back to the start of the loop again, and you get a sudden burst of noise again - the burst of noise could sound a bit like a cymbal being hit; the slowly increasing pitched sound could sound like the start of a cello or reed organ note.
Continuing this line of thought, if there are any strong pitches in the sound being repeated, you may hear a repeating melody. It may be that this corresponds to the musical scale notes we are used to, but even if not, the very fact that it repeats will give it a musical identity.
So there's almost no phenomenon to explain - it's just that repetition is rhythm. By repeating a sound snippet, you can turn an unmusical sound into a musical one. One composer who was very interested in exploring this was Steve Reich - an example of his music is It's Gonna Rain :
We LIKE patterns. Our minds and brains crave for order, for organization. If you write abtruse dodecaphonic music with no tonal basis at all, listeners will strive (often successfully) to find a tonic. Let's not start a fight, but this could help explain why some people are happier with design than with evolution.
The answer to this specific question is much more simple. If you repeatedly loop anything there IS a repeated pattern, certainly of rhythm, probably also of pitch. Who are you to decide anything is "non-music"? :-)