Can you copy and apply the same color changes to different clips?

How do you apply the same color correction to many different clips?

see that when you copy a clip, you're given the option to paste the attributes . . but there is not any option to paste the color correcting attributes you did for one clip to apply to another.

Yes it is. Color Corrector and Color Corrector 3-Way are both filters. When pasting attributes, check the Filters box. Paste Attributes applies the exact filter you applied to the copied clip.