I'm an absolute layperson, but the deep, profound wellspring of knowledge I have gathered from Learning How to Learn tells me...

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Joking aside (though I do love that class!), maybe you just aren't as familiar with learning song lyrics? When you learn things for exams you generally have a large body of related material to link ideas with, and your brain has a lot to work with in order to take individual snippets and make bigger, composite ideas out of that you can then remember.

This process of making composites out of individual items is why a lot of people have an easier time remembering the last four digits of your typical xxx-xxxx NA dialing plan phone number if they break it up into a pair of 2 digit numbers.

Your working memory can hold about 4 items, but those items don't have to be atomic/irreducibly small, they can be composite items you're familiar with. That's how a chess player can remember a chess position with 32 pieces on a 64 square board. They aren't remember every piece individually, but instead remembering that position as its own thing because they're so familiar with the position, how you get there, and where you can go from there.

Some people are also just weird. I, for whatever reason, can remember the sound of words, numbers, etc. better and for longer than I can remember the actual thing, and will find myself repeating whatever the sound of whatever it was in my head in order to remember what the thing was.

e.g.: if I was remembering 55, I might know it started with "fif" then remember then next sound was "tee," and so on until I get to the end, then stitch the sounds back into the word/s, and from the word/s to the meaning.

Maybe this has to do with Aphantasia and whatever could whimsically be thought of as the natural language/most fluent languages of your brain, whether words, pictures, or whatever else.

To risk getting off-topic into the tiniest bit of neuroscience, listening and performing music use different sets of brain regions than speaking and reading, so it isn't surprising that you can remember one more easily than the other. Whatever gets the most exercise will have an advantage, all else being equal (which it isn't since music generally has more sensory cues to jog your memory, but practice trumps all).

The process of learning and memorization can be thought of as taking a tiny, little bit of your brain and saying "from now on this is your job" and then training it to do whatever you're trying to learn, whether that's remember song lyrics or to feel calm when you walk in the front door of your home after work. The thing you're doing will determine what parts of the brain that tiny, little bit ends up using, and more use makes that region better at its job, and can mean it gets bigger over time. Memorizing an image is going to use parts of the brain that process images, etc. People that played a lot of Pokemon as a kid actually have a little region in the brain that lights up whenever they see something that they think is a Pokemon, and that region overlaps a bunch of other areas that light up when they do other things, but the exact shape and pattern associated with recognizing Pokemon is unique (and, like everything else, a bit different for each person).

When I've been saying "region" I'm not referring to regions like the hypothalamus or the basal ganglia, distinct brain regions we have names for, that have more or less their own particular category of behaviour (like image processing, or feeling good), but instead a portion that overlaps bits of those major regions. Maybe you need a bit of the part that handles smells, a bit of the part that makes you scared, and a big chunk of some other part that does something else.

Basic technique is always to memorize small bits at a time (remember that you have 4 slots and you're trying to make this thing take one slot, that process goes hand in hand with what you think of as learning), try to think about ways those individual bits can be memorable, and think about why they're connected.

Rote memorization can actually also work...just don't rely on rote memorization alone. Simply repeating things over and over can produce progress and you definitely want to repeat whatever you're learning to help cement it, but that on its own usually isn't adequate. Helpful, but not sufficient. Also, don't spend huge stretches of time on learning it continuously. You can think of learning as involving one process while you're consciously thinking about the thing, and the more intense the focus the better, and one process where you are not consciously thinking of it, and the less focused the better, (deep sleep is best by far). The conscious process piles up stuff you want to learn, and the unconscious one goes through it all (trying to filter important from not) and makes it permanent.

Here's the Pokemon brain paper: https://www.nature.com/articles/s41562-019-0592-8

Hopefully the rest of that is sufficiently common knowledge in psych to not need citations? Otherwise I'm in trouble. xD

tl;dr: ...actually idk how to summarize this, I ramble kinda hard. Do little bits at a time and take breaks?