I am hearing that the harmonic minor scale has the same notes as the phrygian scale built from the 5th of the harmonic minor scale. But when looking at the A harmonic minor scale I see a G# while the the E phrygian (built from the 5th of the A harmonic minor) has no G sharp---confused!
The 5th mode of Harmonic Minor is called "Phrygian Dominant" but it's different than the "Phrygian" mode of the Major scale. I think that's where you're confused.
So yes Phrygian Dominant has the same notes as Harmonic Minor because it's a mode of it. And it's similar to but slightly different from Phrygian, the 3rd mode of the Major scale. Let's take a look at both scales starting from E and their scale degrees relative to the parallel major scale:
E Phrygian (derived from C Major):
E F G A B C D
1 b2 b3 4 5 b6 b7
E Phrygian Dominant (derived from A Harmonic Minor):
E F G# A B C D
1 b2 3 4 5 b6 b7
If you look closely you'll see that the only difference is that Phrygian Dominant has natural 3rd compared to the Phrygian as derived from the Major scale.
So that's where the similar name comes from. The minor modes are often named because of their similarity to major modes. It makes it easier to remember the scale degrees if you already know the major version (Phrygian) because then you only have to remember the degrees that differ (natural 3rd). So "Phrygian" because of that similarity . And "Dominant" because it's the mode associated with the 5th, or dominant, of Harmonic Minor.
By the way it's called other names as well like "Phrygian #3" or "Mixolydian b9b13", which is maybe even a more helpful way of thinking about it as far as practical usage (ex. playing it over a Dominant 7th chord to either match or create the alterations of those chord extensions).
Phrygian is not a mode of harmonic minor, but of the natural minor scale. The fifth mode of harmonic minor is however called phrygian dominant. Below are both scales, with characteristic tones marked.
The Phrygian scale: 1-b2-b3-4-5-b6-b7
The Phrygian dominant scale: 1-b2-3-4-5-b6-b7
The following resources have more information:
If you think about it, that is impossible. The steps in the harmonic minor are: 1, 2, b3, 4, 5, b6, 7, 8 and there is an augmented 2nd between the b6 and M7 degrees. It is not possible to start from any other degree and get the same arrangement of notes.
In other words, to be a harmonic minor scale, the augmented 2nd must be between the b6 and M7, but if you start from any other degree, the position of the augmented 2nd is no longer between steps 6 and 7.