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.
C4 D4 E4 F4 G4 A4 B4 C5 D5, that's a convention, use it an people will understand your meaning.