No, not really. It may help if we break down what this greeting actually, literally means. Surprise, surprise, it means: "to the seeing (you) again". Notice, there is a "the" in there. That's the "A" in "A viszontlátásra!".
The complete break-down of that long word is this: Viszont - lát-ás - ra
"viszont" means something like "counter-", "re-", "in turn", "again", as in something we want to return, to happen again. Let's say "once again"
"lát" - see
"látás" - seeing, acting as a noun
"-ra" - to, for, onto, something like that.
So, in all, as I say goodbye and extend my hand, I do it in the hopes of seeing you once again soon, and we may even drink "to (our) meeting again", that is, "to THE next time we meet", "A viszontlátásra!".
So, do other languages have similar greetings? Of course they do:
English: See you, see you later - an informal goodbye
Spanish: Hasta la vista - "until the (next) sighting" - Notice the "la"? Kind of the same idea with the "the" or Hungarian "A".
German: Auf Wiedersehen - It means pretty much the same as the Hungarian phrase.
Wieder - Viszont. Sehen - látás. Auf (in the front) - -ra (at the end)
I would not be surprised to learn that the Hungarian phrase is the direct translation of the German one. Hungarian has gained much stuff from German by means of literal translation.
German has one more phrase: "Auf Wiederhören", which is "to the hearing (from you) again." And, surprise again, Hungarian has the exact match: "A viszonthallásra!"
"Hall" - hear. The rest of the word is the same as with seeing.
Back to Hungarian, both "(A) viszontlátásra!" and "(A) viszonthallásra!" are formal greetings. Not like "see you later" at all, which is quite informal. The "hearing" version is used in hearing-only situations: to end telephone conversations. Your favorite radio show host may also use it at the end of the program.
And when to use "Viszontlátásra"? (yes, the "A" is optional). Well, if you greeted somebody with "Jó reggelt", "Jó napot", etc., then you can safely use "Viszontlátásra" to say farewell.
"Viszontlátársa" is a mouthful of a word. So, there is a shortened version: "Viszlát". As in VISZontLÁTásra. This is of course less formal. One would say it to any adult that is not family, friend or close acquaintance. It is widely used.
So, hopefully it is clear why there is no "A" in other greetings. Because it just does not belong there. But here it is an organic part of the original phrase.
You can also use this greeting in reference to a specific meeting, appointment, etc. For example, "See you tomorrow" would be:
"A holnapi viszontlátásra"
"Viszlát" is a shortened form of "Viszontlátásra". Some people also use a medium-length version, "Viszlátásra".
"A" means "the" in English. You can't use it with other greetings (for example, we never say "a jó napot kívánok"), because "kívánok" is a form of the verb "kívánni" that needs an indefinite, not a definite form (you could also say "a jó napot kívánom", but that doesn't really make any sense). However, with "viszontlátásra", it is up to you whether you use the "a" or not, there is no real difference.
I hope I could help, and greetings from Budapest. :)
In Hungarian we make a big difference between formal and informal speech. All "szia", "szervusz", "helló" (note the spelling), "hi" (with English pronounciation!), "ciao" ( and --> "csáó", "csá", "cső", etc.) "szióka", etc. are informal, and latters are more or less part of the slang of the youth. "Viszontlátásra" is the most general, slightly more formal than informal way to say goodbye, and English has the "goodbye" as its closest equivalent. And there is a technical matter, too: if you want to test is in this course in English to Hungarian form, how would you ask it? :D