Why was Mr. Park's first driver very keen on knowing Jessica's location by wanting to drop her at her home?
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.
He was interested in her sexually - she was an attractive young woman, he was an attractive young man. He wanted to spend the extra time with her to get to know her and to impress her by literally going the extra mile in, what I believe the script established was his own time.
He was quite insistent on taking her all the way home until she mentioned she was meeting her boyfriend - and the mention of an existing boyfriend obviously disappointed the driver.
We did not know the social status of the driver - presumably relatively low - and he had no reason doubt the higher status of Jessica the art tutor whom Kim Ki-jung was pretending to be; so I disagree with the other answer that suggests he wanted to check out her status by where she lived.
Obviously part of it is because he wanted to drive her home, but also to confirm her social status by knowing if she lived in a well-off neighborhood. The Park family wants even their servants to be better off than the average person. For example, Mrs Park is hires "Jessica" to be her sons art therapist because she believes her to have been educated in Chicago, and cousin to "Kevin", who is a friend to Da-song's previous tutor Min-hyuk.
Min-hyuk admits at the beginning of the movie that Ki-Woo is better at English than he is:
MIN-HYUK: Ki-Woo, think about it. How many times have you taken the college entrance exam? Twice before your military service and twice after -- a grand total of four times. Grammar, vocabulary, composition, speaking. You’re an English master. Far more qualified as a tutor than I am. Better than those spoiled college brats who drown their brain cells in booze everyday. (link to the script)
He acknowledges that the Park's care just as much as their tutors status as his actual skills.
KI-WOO: That may be true but-- You think the family would accept me? I’m not even a college student.
MIN-HYUK: We’ll embellish a little. You’ll be fine. You’ll have my recommendation.
Min-hyuk knows that the Park's won't look too closely into Ki-woo's (faked) credentials if they come along with his personal recommendation.
Back to the driver though. He was present when Mrs Park introduces "Jessica" to Mr Park. Although the brief introduction doesn't give him too much information to work on, from context clues he can derive that she is well-off enough to be fluent in English and well-connected enough to find her way into the Park's service.
Ki-jung asks to be dropped off at Hyehwa station, which from a ten second glance at it's Wikipedia page, is located in an area that attracts many vibrant, young people. After taking a look at the script, which states
Ki-Jung’s cold demeanor intrigues Yun instead of putting him off. He’s attracted.
I agree that @Mr_Thyroid's answer of the driver being attracted to her does hold some weight. It is likely that the combination of his sexual/romantic interest in her, his general interest in her living situation, and his training as a driver that make him so persistent in wanting to take her all the way home.