I think the issue may be about idiom versus "correct" but unnatural grammar.
A language metaphor may help.
"Put your toys away" would be an idiomatic expression.
"Reposition the toys you played with to the toy storage container" is grammatically correct and has the same meaning, but is unnatural.
If you listened to and were asked to repeat back a series of phrases like those, the idiomatic ones would be much easier to repeat, because the idioms are familiar to you from constant use. In part the idiomatic memory is already there, you just recall it. The unnatural sentence would require new memorization with very little supporting idiomatic patterns.
Twinkle, Twinkle Little Star uses melodic "idioms:" tonic to dominant, dominant to neighbor submediant, descending scale dominant to tonic. Those three short units can be found in lots of other melodies. Importantly when a few of those units are combined it will quickly establish the tonality.
In other words you aren't hearing and remembering - ascend P5 ascend M2 descend M2 descend M2 descend m2 descend M2 descend M2, which is the abstract order of intervals - but rather - ascend tonic to dominant, ascend to submediant, descend scale - which are "idioms" of a major key.
...much harder to remember the SOUND of the tritone or other interval...especially when the latter only consists of two notes.
This sounds like the typical ear training test. I'm suggesting it's difficult, because there is no real musical context, you cannot recognize the interval "idiomatically."
...I'm not asking why we don't "understand" intervals at some sort of "intellectual" level...
Right. People can do this without understanding anything technical about music. Play an ascending major triad and it's easy for a person to sing it back. Ask them to sing an ascending major third and then an ascending minor third and they probably won't understand those terms, even though they understand them with their ears as a musical "idiom."
I'm asking why can't we remember the SOUND of an interval just like we remember the SOUND (and, perhaps, feeling) of the melody of a song we are familiar with.
It's interesting your choice of words "sound" and "feeling." If you consider my suggestion about idiom and context, then "sound" and "feeling" are not fixed abstract qualities. The tritone is a good example. The abstract interval is
A4, but that isn't a "sound" or "feeling." You need context for that. In one context, like diminished seventh chords in a major key, the tritone is discordant. But, in another context like a half diminished chord in an Impressionistic or jazz setting, it's mellow. Or in the dominant seventh chords of blues the tritone is "earthy." Abstract intervals aren't musically expressive until they are put into context. Clearly you would like to "remember" by feeling, but in a ear training test they are usually presented abstractly without context.
Another thing to consider is the memory interaction between word and melody. The two reinforce memorization. In part a song melody is more easily recalled using the words that go with the melody.
This presents a possible strategy for ear training: put intervals into context and use solfege syllables to sing them in lieu of song lyrics. So, don't just sing a tritone, sing the syllables "TI", "FA." Also, play with some extra context. Start with "DO", "MI" and then connect those tones/syllables to those of the tritone "DO", "TI" and "MI", "FA." Then maybe try "DO", "MI" followed by "TI", "FA" and vice versa before finally just singing "TI", "FA" in isolation.