Polaris, the North Star (I) (might be a different star used for navigation) and the Earth (you)
I see you as a circle.
You see me as a point. (unsure)
The earth is typically drawn as a circle, and stars as points. This is the most shaky clue, as from the star's distance the Earth would be invisibly tiny so not really a circle)
And when you need direction
I never disappoint.
Polaris is famously used for navigation, as it doesn't move in the sky
You see me when you're sleeping,
And oft when you awake
Stars are visible at night. They are occasionally visible during the day during a solar eclipse
(Even in the daytime
I sit and burn and bake).
Stars still shine during the day, it's just too bright to see them.
Your friends and you are whirling.
Is that really you?
Planets in the solar system spin both on their axes and around the sun
Yes! Of course! I recognize
That swirling bit of blue.
Earth is known as the blue planet, and the clouds are swirling in photos from space
Anagram for Umbrella, misspelled
From JS1 in the comments: polaris is an anagram of parisol(parasol), which is a type of umbrella