However, I have a book about scales for guitar where I found this Persian scale which, to me, sounds exactly like the Double harmonic one:
Am I missing something?
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They sound nearly exactly the same. There is a slight difference, in that the 5 of the Persian is a ♭5, whereas the 5 in the double hamonic is P5. Playing a melody which didn't have a 5 (quite rare!), it would be impossible to say which scale is being used out of the two.
The one posted has a P5, so can't be the Persian scale. And seems to go awry in its middle. Sometimes, there are inaccuracies on the net - particularly on, it seems, guitar sites...