All right, you got some excellent answers, let's check and extend them!
So first of all, Szakál (currently written as szakáll, names often use archaic forms, e.g. Kovács-Kováts, Kis-Kiss, Tót-Tóth, Horvát-Horváth) means beard, that was already established, also this form uses the Hungarian alphabet ("SZ" is quite telling). The word itself exists in the Turkish language as well, but I'm not sure where it came from. Hungarians met with Turks before they came to the Carpathian basin, and there was also the 150 year long occupation. They created their version of the Latin alphabet based on Hungarian in the later period, so the direction was two-way. Nevertheless, there are many common words.
I've found this in an online ethimology dictionary (link in Hungarian). My rough translation:
Ótörök eredetű: ujgur, türk, tatár, kun szakal, oszmán sakal (‘szakáll’). A szóvégi al ⇨ áll nyúlás a magyarban következett be.
Old(ancient) turkish origin: uyghur, turk, tatar, kun: szakal, osman sakal (‘szakáll’). The elongation at the end of the word al ⇨ áll happened in Hungarian.
I think your Persian theory is a bit far-fetched. After all, "Úr" means "Lord", "Uruk" means "their Lord", and although Isthar is not a Hungarian word, however "Isten" is (means "God"). Despite that, noone suggested we came from Mesopotamia or something, coincidences are a thing.
That "personal characteristic" matches this perfectly. This is very common, Nagy (Big) and Kis (in various forms) (Small) are among the most common names (I think only after Kovács (Smith)). Balog is an old term for left-handed. But names about hair are common as well. Kopasz (Bald), Szőke (Blonde), Bajusz (Mustache), Hajas (Hairy - the hair on the top of the head, not the other one). So Szakál(l) fits into this line well.
Szekély is probably a misspelling, the tribe in Transylvania is called Székely, and yes, that's unrelated.
I've had a coworker whose name was Szakál, he wasn't Jewish. Also, he was the only one among the Android developers who didn't have beard. :) (but again, it isn't a rare name, so one example doesn't really matter)
Given names are a lot easier. The Academy has a language group, they keep up-to-date two lists: one with girls' names, the other for boys (there is no third yet, the progressive party fell apart). Parents can choose names from these lists, or - mostly for minorities - get a permit to choose something from their culture. There are also books with all the allowed given names and origins, etc. Vladimír is on the list (written like this). As far as I know, it came from Russia, so I agree.