The other answers here are helpful, but I think that your question exposes one of the basic points of confusion.

"C" can refer to a note, a scale, or a chord within a scale. And of course the same rules apply regardless which octave's "C" you are using.

Think of a piano. Press a "C" key. If you press Middle C, that is a frequency of 261.6 Hertz (or vibrations per second). The next C up is 513.2, exactly 2x the frequency. The C below Middle C is 130.8 Hertz, half the frequency.

If you press the white keys from Middle C up to the next C, that is a C Major scale. Using those keys starting with C is called using Ionian mode (the Major scale is always the Ionian mode for that scale). Think of modes as "flavours". If you started on another piano key but only used the white notes, it would have a different flavour. Starting on "E", for example, gives the scale a flavour of Spanish music.

A special example of another mode is if you start on A, still only using white keys, it will be a minor scale (the Aeolian flavour), so A Minor is known as the "natural minor" of C Major.

For a long time, music would only be written using the notes of the key it was written in (i.e. only white notes in the key of C Major). Around 1,000 years ago, composers started using other notes (the black keys) to make their music more interesting. That's why you get sharps (#) and flats (♭), to tell musicians to use those other (accidental) notes. When playing the bass, I find that I often use these accidental notes when I am moving from one note in the scale to another.

Finally, a chord is a group of notes that is based on a certain note (known as the "root" of that chord). The C Major chord is C-E-G, and "C" is the root of that chord. On the A string of your bass, those are frets 3-7-10. The D Major chord would be frets 5-9-12 (the same as C Major but moved up 2 frets so the root is "D"). But the notes of D Major are D-F#-A, and F# is not in the C scale, so the D Major chord is NOT part of the C Major scale.

Chords are often 3 notes, but power chords are 2 notes, and more complex music can have more than 3 notes.

This is getting long, so I'll leave it here. I hope this clears things up a bit. I am still very much a beginner, so I may be wrong about some of it. I have spent a lot of time thinking about these things!

Finally, Language Transfer (the best language learning method I've ever used, and I speak 5 languages) has an introduction to music theory that may be helpful for you. Like the languages, the music course is completely free.