What does "vergeben" relationship status mean?

"Vergeben" literally means something like "given up."

In a "dating" context, the English expression is "taken." That is, when you have been "taken" by someone, you have"given up" yourself to someone to that person.

In the more usual context, if you have "given up" your grudge at someone's causing you offense, you have "forgiven" them.

Geben translates to "give", but translating vergeben to "forgiven" is a false friend in this context (but possible in other contexts).

vergeben:
Bedeutungen

etw. (an jmdn.) weggeben, verteilen
umgangssprachlich: zwei Töchter der Familie waren schon vergeben (= verheiratet, verlobt)
–– colloquially: two daughters of the family were already taken (= married, engaged)

As a figure of speech it's still very similar to see this connection:
Have you seen a marriage ceremony? The father gives away the bride.

“given away” could be Past Participle give away verb
vergeben v
On the occasion of its anniversary, the company gave away gift vouchers.
–– Zum Anlass ihres Jubiläums vergab die Firma Geschenkgutscheine.
_{Linguee Dictionary: given away}

So, depending on perspective vergeben could be read as "already taken" or "already given away".

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The practical explanation above is rooted in the theoretical explanation provided by linguistics when looking into the etymologies:

Geben and give are as much cognates as vergeben and forgiven:

German cognate | Meaning of German cognate | English cognate | Proto-Germanic root
geben, gab, (ge)geben | to give | give, gave, given | *gebaną
^{WP: Loanwords from Old Norse}

forgive (v.)

Old English forgiefan "give, grant, allow; remit (a debt), pardon (an offense)," also "give up" and "give in marriage" (past tense forgeaf, past participle forgifen); from for-, here probably "completely," + giefan "to give" (from PIE root *ghabh- "to give or receive").

The sense of "to give up desire or power to punish" (late Old English) is from use of such a compound as a Germanic loan-translation of Vulgar Latin *perdonare (Old Saxon fargeban, Dutch vergeven, German vergeben "to forgive," Gothic fragiban "to grant;" and see pardon (n.)). Related: Forgave; forgiven; forgiving.
^{Etymology of forgive}

vergeben Vb. ‘weggeben, austeilen, verschenken, verzeihen’, meist reflexiv ‘seinem Ansehen schaden’, nur reflexiv ‘sich beim Austeilen (der Spielkarten) irren, falsch geben’, ahd. firgeban ‘vergeben, schenken, einräumen’ (8. Jh.), mhd. vergeben;
^{DWDS: geben/Etymologie}

That means that both words share the exact same root but the English word forgiven has lost a part of the original meaning ("to give away"/"to be given away") while the German word vergeben retained that meaning and expanded it to encompass not only married people (specifically women) but now means any sex and gender committed in a relationship.

While πάντα ῥεῖ has already answered the question, let me explain why your assumption wasn't a bad one.

Does it mean that a person is forgiven?

In German-like languages, it could've been literally translated like that and still work. However, English is not a German-like language. In Dutch, however and for example, it would've worked. "Vergeben" would translate to "vergeven", which either means "forgiven" or "already occupied/taken/given away", but also "awarded", "handed out" or "assigned". It can be translated one-on-one regardless of context (although it looks archaic in some of those cases).

It's one of those words with multiple meanings which can translate directly to another language with the same meanings or a language where the meanings are split up. There's probably a term for it.

The vocabulary and grammatical similarity of German to English are simply too low to reliably make literal translations like that.

Vergeben in relationship status means that those people currently are in a relation already.

Similar to the English, taken.

It has nothing to do with the English forgiving, but simply means these are already committed into an existing relationship.

In German vergeben is used for several things depending on context, like

• Der Platz ist bereits vergeben

  That seat is already occupied (taken)

• Wir vergeben ihm, da er es nicht besser wusste

  We forgive him, since he didn't know better