How is "officially" used in a non-literal way?

In this colloquial usage, officially is roughly synonymous with definitely or indisputably.

The origin of this sense of the word lies in the fact that many terms that have vague dictionary definitions also have very precise definitions in some laws or regulations. Consider, for example, poor. If one doesn't have much money, but is not completely penniless, it may be debatable whether one is poor. However, if one's income is below some precise amount that the government has set as the poverty level, then one may be said to be 'officially' poor. Or consider the word dependant. If one is a young adult with a part-time job, who receives some financial help from one's parents, it may be debatable whether one is a dependant of the parents. One may, however, be said to be 'officially' a dependant if one's finances fit some precise criteria specified in the taxation code. In all these cases officially carries an implication that the matter is definite and indisputable. That makes officially an apt word to be used as a playful synonym for definitely or indisputably, even when there is nothing literally official about the matter.

I agree with your assessment. The figurative usage expresses the culmination of a process by which a person crosses some kind of emotional threshold with respect to an issue or situation.

The use of officially is borrowed for this because official things are normally declared by a governing agency of some sort

officially
2. Authorized by a proper authority; authoritative
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and in our own minds we are setting up our own governing body (our feeling of having had enough, usually, of whatever it is) and making a declaration to all who may hear. Compare it in spirit with the paid notices that appear in newspapers whereby someone's spouse declares they are "no longer responsible for so-and-so's debts or behavior"—that sort of thing. It says to anyone listening that a line has finally been crossed.