Also, i have my own theory in which this surname may have a connection
to ancient tribes Scythians whereas Persians called them "Saka"
(possible Szaka - Szakal?).
Unlikely, because surnames only became a convention in Central Europe in the middle of the last millennium, and the peoples of Hungary, like most peoples, took their current surnames even more recently, based on:
their occupation (Lakatos, Pásztor, Kovács, Szabó)
their ethnicity as understood in the 19th or 20th century (Görög, Nemet, Tót, Székely, Örmény, Oláh, Rácz) OR country of origin
their city or region of origin (Kecskeméti, Visontai)
the Hungarian rendering of their original non-Hungarian surname (Liszt, Kossuth, Dóczi)
the original rendering of their original non-Hungarian surname (Wigner, Neumann)
the translation of their original non-Hungarian surname (Hidegkuti)
the personal name of their father or ancestor (Ádám, Lázár, Benes)
some personal characteristic (Féher, Nagy)
See https://en.wikipedia.org/wiki/List_of_most_common_surnames_in_Europe#Hungary and https://en.wikipedia.org/wiki/Magyarization#The_Magyarization_of_personal_names
Often what happened was that people chose a name with a meaning in the new bureaucratic language because it sounded similar to their original name, by a very loose definition of similar, for example Kohn was Magyarised to Kun, Purczeld to Puskás, Günszberg to Gábor, Engländer to Erdős.
Therefore, the name in the case of a family could have an actual origin distinct from the origin of the current Magyarised surface form.
So why would somebody in Hungary have chosen Szakal (Sakal in other orthographies) in the 18th, 19th or 20th century?
According to ancestry.com, Sakal is a Jewish (Sephardic): occupational name for someone who polished wood, gems, metals, or armor, from Arabic ?saqqal ‘polisher’. Hungarian (Szakál): descriptive nickname from Hungarian szakál ‘beard’, Turkish sakal.
(For those who do not know, Szakál in Hungarian orthography would be Sakal in many other orthographies.)
It also lists a Szalmon Sakal, which is Polish orthography. (The same name in Hungarian would be something like Salmon Szakal.) Given that there were very very few Sephardim north of Hungary, the given etymologies seem not to apply.
Other ideas: In Latin sacal meant Egyptian amber. Szakál is also the Hungarian name of the Săcal River in Transylvania.