I'm a total newbie. And I'll pretend there are no black notes so it is easier for me to explain.
CDEFGABḈḊ (where Ḉ and Ḋ are one octave above C and D)
When a Musician says, "oh, C and Ḉ are the same note" (Sound or pitch? wise), is it because music theory says so or is there a scientific reason? (for example, like Li and Na have same-ish properties in the Periodic table.)
We say that C and Ḉ are "the same note" because their frequencies (rates of vibration) are in a 1:2 ratio. In other words, the sound waves of Ḉ vibrate twice as fast as those of C, and the waves of the pitch an octave above Ḉ vibrate at twice the rate of those of Ḉ (four times the rate of those of C).
In music theory, we distinguish between pitch and pitch class. A pitch is any given musical sound; C and Ḉ are both separate pitches. A pitch class, however, is the collection of musical sounds that all have the same name. Thus C and Ḉ are both members of pitch-class C, even though they are separated by octave. Similarly, D♭ and Ḋ♭ are all members of pitch-class D♭.
Another similar term is that of octave equivalence. At least in tonal music, because of the how the vibrations align, we treat C and Ḉ as "equivalent" from this stand point; hence the use of the term pitch class, which assumes equivalence across all octaves.
Music theory says that notes with the same name are "the same note" even though they are obviously different pitches. When played together, it can sound like a single pitch even though there are two or more.
By designating C and Ḉ as "the same", it allows for scales, for example to be seven pitches that repeat rather than infinite series. It also allows for chords to be defined consistently no matter which octave the notes happen to be in.
To the degree that there is a scientific basis, it's that given a particular pitch, its first overtone is one octave higher (and further octaves are also within the overtone series). Thus, when we hear C, we're also hearing Ḉ (as well as other overtones), but we perceive it just as the single pitch C.
is it because music theory says so or is there a scientific reason? (for example, like Li and Na have same-ish properties in the Periodic table.)
Well music theory does say so, but you can empirically determine this yourself by listening to octaves. They sound very similar to each other.
I agree with Theodore's comment, the periodic table comparison is pretty good. Elements have an electron shell with a certain shape, this determines how they react with other elements (and why Li and Na are in the same "group"/column.) All notes have an overtone series, and many people believe the way two notes sound together has everything to do with how their overtones overlap.
In the case of the octave: if our fundamental pitch is 100Hz, the sounds produced would be 100Hz, 200Hz, 300Hz, 400Hz, 500Hz, 600Hz, etc. You hear all of these but interpret them as a single pitch. If you played the octave, the sound would be 200Hz, 400Hz, 600Hz, etc. You can see that the lower octave's overtones contain every one of the higher octaves overtones. When played at the same time, all you get is a change in amplitude of frequencies that are already present.
Back to the periodic table, Halogens and Alkali metals react a certain way because of how their electron shells complement each other. The notes C and E react a certain way because of how their overtones complement each other. This reaction creates chemical compounds, intervals, chords, and the reaction is similar regardless of octave.
Don't completely disregard octave. We season our food with NaCl, not KBr despite both compounds being made of the same groups. Similarly, we don't usually write basslines in the range of C5.
