Short answer: I wouldn't describe them as the same but they do have the same function. They are both dissonant chords that desire resolution and both resolve to the same place with very similar voice leading.
We can make this a little more simple to start. E7 is V of A; G#° is vii° of A. Both of these chords want to resolve to A with G# being the leading tone that wants to resolve to the tonic, unless you're playing Jazz, in which case it may be held as the major 7 of an Amaj7 or step down to b7 for an A-7. D is the 7 of E7 or the b5 of G#° and will want to resolve down by step.
Once we add in the b9, the texture changes a little but we have the same functionality. If we utilize the half-whole/whole-half scale, we can then shuffle around our concept of resolution. This scale will create 4 Dominant chords, all a minor third or tritone apart. For E7 we could use the E half-whole scale (E,F,F##,G#,A#,B,C#,D). The 4 Dominant chords that can be derived from the scale are E7b9, F##7b9 (G7b9), A#7b9 (Bb7b9), and C#7b9. The reason this works is because all of the 4 contain the ever important tritone that desires resolution. For E and Bb (Bb is also the standard tritone sub), the tritone falls on the 3 and 7 of the chord; for G7 and C#7, the tritone falls on the 5 and b9. This allows us to have the expected voice leading and resolution for a standard Dominant chord with any of those 4 choices. In the same regard, a fully diminished 7 chord can use the whole-half scale, which will create 4 fully diminished 7 chords (it actually creates 8 fully diminished chords but only the 4 are typically used within the scale). The 4 notes that make up all of these chords are the same as the 3,5,7,b9 of the Dominant chords outlined above. The same way that we are able to state that E7 and G#° have the same function, we can say that these diminished chords have the same function. So you can actually use E7b9, F##7b9 (G7b9), A#7b9 (Bb7b9), C#7b9, G#°7, B°7, D°7, or F°7 to fulfill the same function. This is more commonly known/understood for fully diminished 7 chords, as they are symmetrical (all minor 3rds).
So I wouldn't say that E7b9 and G#°7 are the same chord but they can fulfill the same function, along with the bunch of other chords I mentioned. The thing to keep in mind when thinking of things this way is where this is being applied. If you're composing or arranging a Classical piece, the treatment of these chords will be a little different and whether or not the texture appropriately applies to your piece will probably be very specific to each piece. In Jazz, this sort of approach is very common but it may not be as appropriate in certain subgenres, such as Big Band Swing or Modal Minimalism. You'll probably find that this approach doesn't work so well in Rock or Pop music, largely because of the diatonic nature of these genres but also the standard chord voicings. Like any other musical device, your taste and ear should be used to make the determination as to whether or not this sort of substitution approach should be used.