Other great answers here -- I'll add my $.02, well, just because.
Ham here as well (K2KD). I had most of my experiences as a kid (6th - 8th grade), had an HF rig in my bedroom, and made my way through Extra class (which, I think, required a 20 WPM morse test).
I'd liken hearing common CW phrases not so much as hearing "the phrase" in its entirety / natively, but more akin to listening to a song or tune you're familiar with. Your brain knows what the next word / note is going to be, so that's queued up in your head (and why you can hum along with a tune you know).
Same with morse code -- you'll start to hear a familiar pattern and you'll just know how it's going to continue. Things like "CQ" were VERY common to hear and you always knew that pattern (-.-. --.-). Or signing off, you knew when someone was starting to say 73 (--... ...--) so you'd end up "singing along" with the words to the song you knew (metaphorically-speaking).
I was quite experienced with it and even now, 20 years later, I still know the alphabet. But I'd never say I got to true language-less fluency -- the code always translated into letters for me (either written or in my head), except in the cases above (I can hear a CQ now and not think "CQ" but instead hear it as the universal beacon call...da dit da dit..da da dit da......da dit da dit..da da dit da...)
Good memories - glad you're giving it a try!