The "Milky Way" is a band of light caused by the light of millions of stars in the spiral arms closest to our position. Individual stars in those arms are too far to be seen with the naked eye, but the total amount of light they give out can be picked up by the eyes; the brain interprets it as a diffuse band of light (relatively faint).
The brightest part of the Milky Way (as seen from Earth's surface) is towards the centre of the Galaxy, in the direction of the constellation Sagittarius, well south (30 degrees) of the celestial equator -- therefore NEVER seen from the North pole.
However, the bands are visible on either side of the centre. From the North pole, we cannot see the southern portion.
The part that comes up from Sagittarius crosses the celestial equator in the constellation Aquila (the eagle) and continues into Cygnus the Swan (its main star Deneb is at declination 45 N) and remains visible up to the southern parts of Cassiopeia (near declination 60 N). It then arcs back down towards Perseus where it gets fainter and fainter (we are then looking in the direction opposite to the Galactic centre).
That portion of the Milky Way (Aquila to Perseus) is ALWAYS above the horizon for an observer at the North Pole. However, it is not visible during the "day" nor during the Civil and Nautical twilights (when the Sun is less than 12 degrees below the horizon). That period goes roughly from Feb. 15 to Oct. 25.
This leaves a period of 112 days (Oct. 26 to Feb. 14) during which the sky could be dark enough to see the Milky Way.
Except that when the Moon is bright and above the horizon, its light is often enough to prevent you from seeing the faint Milky Way. And, unfortunately, during the winter months, the Moon will be above the horizon from roughly First Quarter to Last Quarter (this period includes the Full Moon). Like everything else in the sky, as seen from the North Pole, the Moon would not set during its bright two-week period.
Every lunar month, you would have to wait for the period from Last Quarter to First Quarter (including New Moon), when the Moon is below the horizon, to see the Milky Way. This leaves you a total of (let's say, generously) 60 to 70 days per year during whihc, IF THE WEATHER allows you, you could see the Milky Way from the North Pole.
Standing directly at the pole, the Milky Way would arc up to 60 degrees above the horizon.
For almost 2000 years, The Milky Way was (and still is) the name for the bands of light we see, caused by the light of millions of stars located in the nearest parts of the Galaxy's spiral arms. For a while, after astronomers understood that all the stars we see (in addition to the Milky Way) were part of a large unit, that unit was called the Galaxy (capital G). It was thought to be the only galaxy in the universe. In fact, the word "universe" was often taken to be a synonym of Galaxy.
The discovery of "other galaxies" is relatively recent (a little more than a century). These other galaxies were given names based on where they were seen (the name of the constellation, such as the Andromeda galaxy, seen in the constellation of Andromeda, the Triangulum galaxy in the constellation called Triangulum).
As more were discovered, they got names based on how they looked in a telescope (the Sombrero galaxy, the Owl galaxy). And we now know of so many that we give them catalog numbers (NGC 224 is the 224th entry in the New General Catalog -- it is the entry for the Andromeda galaxy). In that context, the word galaxy is written with a small "g". Our Galaxy's name was simply the Galaxy, written with a capital "G".
Then Americans got into the nasty habit of slapping capital letters where they don't belong and (much worst) removing them where they do belong.
This left American readers with the impression that our Galaxy had no name (when it did), leading people to call it the Milky Way Galaxy (which is not totally wrong, since it is the galaxy that contains the Milky Way).
If you want to use that name for our Galaxy, then you must use the word Galaxy in there. The other one is not called "the Andromeda"; it is called the Andromeda galaxy (to distinguish it from the Andromeda constellation, for example).
"Milky Way" by itself is still the name of the band of light.
Observer's Handbook of theRoyal Astronomical Society of Canada
Sky Atlas 2000.0 (Wil Tirion)
6 trips in the Arctic.