I don't believe he was 'checking on her social status'. I believe he was just being conscientious in wanting to take her door to door. The script also indicates that he is attracted to her, therefore might be doing this to be nice to her:
YUN
Where do you live, Ms. Jessica? I
might as well just drive you
home--
KI-JUNG
It’s fine. Just drop me off at
Hyehwa Station. Thank you.
Ki-Jung’s cold demeanor intrigues Yun instead of putting him
off. He’s attracted. He tries again.
YUN
Doesn’t matter if it’s far. I’m
done for the day--
KI-JUNG
I’ll get off at Hyehwa.
Yun looks out the window.
YUN
It looks like it’s about to rain.
I bet the Mercedes is way better
than the subway--
KI-JUNG
(cutting him off)
No. I’m supposed to meet my
boyfriend at the station.
YUN
I see.
Yun’s smile disappears.
Ki-Jung ("Jessica") on the other hand didn't want the driver to see the actual neighborhood and place she lives in. This may be because she was being cautious and not wanting to reveal her true neighborhood to someone connected with the Park family.