Well, it is most often a statement about a time available for an appointment/anything social, really, but less specific than "tid" which would only imply an appointment with a doctor/therapist etc. for a shorter time. Could be said like:
"Hvornår kan du mødes med dronningen?" "Jeg har en mulig dato i morgen, (den 6. juni)"
"When are you able to meet the Queen?" "I have a possible date tomorrow (June 6th)"
Except we wouldn't say that in English - a "date" in English in the above context means a romantic encounter.
In your example with the Queen, if you said "I have a possible date tomorrow", the answer would be "Lucky you - how about the day after?"
It's confusing to use the English word "date" as a translation here, but it's made more complicated by the fact that it doesn't look like there's a direct translation of "dato".
The best translation of the above sentence into comprehensible English would be "Tomorrow's a possibility for me".