I've read this sentence in a text which in the tense of situation is present, so why the past tense of "get" has been used? And does that mean "I have a date with..."?
Some more context would be nice, but generally, yes- it does mean I have a date with [...]
Consider this example conversation between two people:
A- Hey, you wanna come over and study tonight?
B- Sorry man, I got this party tonight.
Person B says there is a party tonight which he has to attend. Technically, yes the past tense of get is used. But, it's an informal way of saying there is a party tonight. You could also think of the sentence like this:
I have got this party tonight.
OR less informally
I have got (to attend) this party tonight.
All of them imply the same thing!
Got also has nuanced meanings. To use the example of the question's title: "I got a date with..."
It could mean:
I have succeeded in getting that popular girl, who is much too intelligent and pretty to be dating the likes of me, to go out to dinner with me on Friday night.
"I've got a date with ..."
would mean simply "I have a date with ...."
"I got a seat on the flight" would mean
I managed to get one of the last open seats on the flight.
The two phrases mean different things.
I got a date with Janet.
Let's say you have been trying to get Janet to go on a date with you. Finally she agrees. You tell your friend "I got a date with Janet". This is in the past (the agreement to the date).
I have a date with Janet.
Now we are talking about the future. Your friend asks you to go to the movies. However you can't go because "I have a date with Janet". This (the date) is happening in the future.
The next day (after the date is over) you might tell someone:
I had a date with Janet.
Now the date is in the past.