Great Question! Typically when tuning, you want to use a pitch that is either a.) within the range of the instrument being tuned, or b.) a pitch that is within a comfortable singing range for you.
Using a pitch within the range of the instrument is important if you want to directly compare two pitches, such as checking for a "wobble" in the sound caused when two identical octave pitches, such as two A4s, are not exactly in tune. My guitarist usually checks his strings after using a tuner using harmonics to make sure the strings are also in tune relative to each other, listening specifically for that "wobble."
Tuning using an octave within your singing range is also useful, even if it's not the same octave that the instrument will be tuned to. It is much easier to differentiate between pitches in the middle and upper registers of a piano than in lower pitches. I find having a higher reference pitch which I can "sing down" from helps me find pitches easier. When tuning timpani, I'll often sing my reference pitch into the drum to hear how much it resonates, as it will resonate more when both it and I are in tune. This is often where you'll get the most mileage out of your college-level ear training classes, as you can easily figure out if a pitch is in tune.
The reason orchestras use A 440 or other As in that range is because it is a happy medium from which all of the different instruments can compare their pitches, and you'll often hear many wind instruments tuning multiple octaves of their instrument instead of just their lowest pitch.
TL;DR: If you have good ear training, tuning to a different octave likely won't make a big difference in your sound. If you have trouble hearing the difference, tuning to the same octave and listening for the "wobble" in the pitch will tell you if you're out of tune. The most important thing when tuning is to make sure pitches are in tune relative to each other!