Is this a date like an encounter, or a date like a certain day in the future?
"Dato" is only a date as in a certain calendar day, like March 3rd . A date as in a meetup is just "date" and a loanword from English. Technically, it would be "stævnemøde", but no one under the age of 60 uses that word, ever, and you will get weird looks if you do.
So what does this sentence mean then? If it's not a date as a romantic rendezvous (that would be date) or a statement about a time available for an appointment (tid) is the whole concept of tomorrow being assigned a date in a calendar called into question?
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".
I guess "availability" would be the best choice, but it's definitely not as simple a translation as just replacing the word.
For me, given the right context, many (though not all) of these seemingly odd DL sentences are possible. Sometimes you need to do a little mental gymnastics first.-- I know you're busy. Get your calendar out. What dates are you free to get together? ...
Opening could also be used in English since we're referring to scheduling
I keep wanting to translate 'i morgen' as 'in the morning' which would technically be tomorrow, but duolingo doesn't accept it. Is it a really big difference, or is duolingo being too picky?