Don't think about the abstract notes as only "letters" — with practice, your brain will create an ever-stronger permanently hardwired physical mapping
which unconsciously transforms interval spacings between notes as noted in a score and the literal physical distances on the piano keyboard itself.
I have been practicing piano since ~2004 (age 13)(; I was immediately hooked/obsessed!).
No I definitely remember learning "FACE" and all that, but that's not how I process sight-reading. It's just like you say as with you and guitar -- it's a physical (touch/sensory/3D) mental mapping. I look at middle c and my hand (depends on context which hand!), reflexively, moves in anticipation of needing to strike middle C to the middle of the keyboard. So you sort of must always be a bit ahead of your hands' actual instantaneous motions - but don't think about that (the movement of fingers itself - this can easily cause you to crash/blank in the middle of otherwise smooth playing).
EDIT: All this and what I'm about to say said, it is nonetheless still important of course to know which notes are which "letters"/tones (of which there are only 7, plus 14, - 9 enharmonics [i.e., f is f, not e-sharp, c is c, not b-sharp, while c-sharp is indeed c-sharp (though also d-flat, enharmonically)], accidentals = 7 + 5 = 12 total tones); some people claim to have synesthesia and associate distinct mental colors with these 12 different tones, I know I do that to some partial degree. Ultimately yes I do immediately see notes on the pages, associate them with the keys on the piano, and think "ah thats 'd' 'f#' 'a' 'g' - a major triad followed by a fourth relative to that tonic", and this is where the music theory knowledge comes in. But more intuitively, I think (it's hardly thinking; as I'll explain in more detail below), or rather see, the notes themselves on the piano keyboard and just go straight to them.
Practice until the hand/finger movement becomes autonomic and completely reflexively built into your neural map-mind. In your brain think of the music notation as being a projection that extends the length of keyboard, from left to right, lowest to highest, in 88 equal chromatic half-steps. Reading "ledger lines" (lines extending below or above the main five of the staff) can be a pain (although really you should eventually have those perfectly mentally thoughtlessly mapped, physically, too). However most of the time, though not always, a composer will spare you the difficulty and use what are called ottava (i.e., "8va" for play this +1 octave, "8vb" -1 octave, and technically "15ma" for +2 octaves higher but that is extraordinarily rarely used) - nonetheless you may as well learn all the notes of the ledger lines as well, in creating this bespoke neural spatiotemporal mapping autonomy for reading scored piano music: i.e., where note [on the page → then goes through the mind thoughtlessly due to hours of practice having sufficiently reinforced connections amounting to what is probably quadrillions of synapses (⁂)] flows to hand to finger to note [now the literal note — the key on the keyboard!], in whichever extreme octave it may be...and of course all in between especially so.
But the best piano music really does utilize the full range so you'll need to be comfy in all of the 7 total octaves. It's all the same thing: 12 notes, repeated 7 times (as there are hours in half a day, and days in a week; these numbers are important and not coincidentally meaningful in music as in religion/spirituality/science/etc.).
(12 * 7) + 3 + 1 = 84 + 4 = 88
It's 7 twelve-tone octaves plus three extra notes in the bass giving you the full relative minor of C (A0 octave) and C8 to cap things off, in, basic, perfection...
Notice for example that octaves are always opposite in their conjoining with a line (be it staff [so for piano that'd be the treble (𝄞) or bass (𝄢) clef staves, of course] or ledger): one of the two frequencies [of the same note "letter"; e.g., C1<->C2 octave] will always be in between two lines, and the other on a line (e.g. C2 intersects the second line beneath the bass clef staff's five standard lines, while its half-frequency lower partner, 12 half-steps below, will be noted in the position tucked beneath the fifth ledger line below bass clef※). You will quickly pick up/literally neurologically create a physical spatial sense of these intervals and autonomically project the spacing of these intervals to the physical distances on the piano itself and the hands are best left doing their thing without "overthinking" about them -- much like skateboarding, or riding a bike - you just do it and your brain is good at that; the bigger picture direction of where you're actually going is what you think about.
Main figure attempting to help illustrate what I'm saying
(Click to expand to larger version, preferably while holding down cmd/ctrl so it opens in a new tab 😉)
A ROYGBIV-Coded Map of the 88 Keys of the Modern Piano, Notated in 11 Equal Measures
Here is a link to MuseScore where you can see/hear the above figure I made "played" aloud*, if curious: https://musescore.com/john_collins/music-stackexchange-com-questions-73730-john-collins
*Little Tangent Commentary on LH/RH Piano Notations (re the Above MuseScore Snippet)
It's not a horrible exercise to practice either, in itself (switching hands correspondingly with the clef switch, so, switching from LH† to RH‡ precisely at middle C).
† [confusingly often notated in [older] scores as "m.g." (French main gauche, left hand) or "m.s." (Italian/Latin mano sinistra)]
‡ [likewise, except by coincidence the French and Italian/Latin end up both being abbreviated "m.d." — main droit and mano destra/dextra, respectively]
Tip for remembering this confusing situation:
So, remember that "m.d." always means right hand (I think "dextrus" as in dexterity and for some reason this reminds me of the right hand, I guess because most people are "right-handed" and thus have more "dexterity" in the RH [as pianists we of course give the LH much more focus 🙂]); and then the others you'll ever encounter will be "m.g." or "m.s."
I think "ok not 'm.d.' so must be LH, and I also remember that sinistra means sinister and, fun fact, for centuries if not millennia, at least in Western cultures, everything "left" (e.g. the hand/side of the body) was considered evil for some bizarre reason; undoubtedly very fascinatingly related directly to us having cleanly separated left and right brain hemispheres; but, alas, such full elucidation of those brain details will continue to elude our understanding probably for quite some time.
A concluding thought in regards to improvisation, its importance, & the benefits provided to it from being well-versed with the full technicalities of the art of scoring music / sight-reading
As a final concluding remark, I would implore anyone new to learning piano (or actually at any stage), to ponder the following thought experiment (which builds heavily upon all I've alluded to here):
What if you were to just start playing random notes with both hands, improvising, and, although obviously it would not sound like good (let alone great) "music" at first, if you kept at it, and literally "live", in-sync, used your mind as a feedback corrector.. through continual re-iterations fine-tuning certain internal weights of biases for playing with your hands particular patterns of harmonic/melodic/note combinations (e.g., just playing [boldly] and experimenting, essentially, without knowing what overall sound(s) will result [though applying your constant best predictions, based both on previous experience as well as knowledge of music theory], and reacting, as such, "ahhh that sequence or combo sounded horrible, play less of those in future" + "ooo that sounded interesting, play more of sequences like that"), would you not, over enough time, develop a profound ability to improvise and perhaps even then compose great music?
Improvisation, is, IMHO, the most important of all piano practice methods. And the beautiful organization of modern music notation we have inherited implies equally beautiful laws, patterns, principles, etc., which will help aid you in your mind, as you attempt to navigate the entire (endless, as Bernstein, and many others, have been persistent to point out) possible space which is human-perceived music.