Yes the 1 mile vision is active all the time.
This is the 3rd level benefit :
Eagle. While you’re raging and aren’t wearing heavy armor, other
creatures have disadvantage on opportunity attack rolls against you,
and you can use the Dash action as a bonus action on your turn. The
spirit of the eagle makes you into a predator who can weave through
the fray with ease.
It specify that the bonus is taking effect while you're raging.
This is the 6th level benefit :
Eagle. You gain the eyesight of an eagle. You can see up to 1 mile away with
no difficulty, able to discern even fine details as though looking at
something no more than 100 feet away from you. Additionally, dim light
doesn’t impose disadvantage on your Wisdom (Perception) checks.
It does not require you to rage to gain the benefit so the enhanced vision is passive
For the spyglass I would say that you would see the exact same thing with or without the Eagle vision, as your sharp vision stops at the beginning of the looking end of the spyglass.