Having a person join you on your trip is not unusual. If there are days at the start or the end for you to explore the area, it is fine to do so with this extra person.
The key is to make it painless for the company, and not cost them any extra money or time.
If you aren't changing the departure date they don't have to know. You don't have to tell them. One thing to consider could be the size of the room. If the hotel room only has a single bed, then you will need an extra room. The person you are traveling with should either meet you after the interview day or be comfortable being on their own.
If you are extending the trip beyond the two days they give you for sightseeing The important thing is to change the departure flight yourself. Because they are willing to pay for two extra nights in the hotel, you should have the bill generated after the two extra days so that they don't see the rest of your stay. Many hotels can do this without any problem.
If you are concerned that the two days will not be free, then ask them. Tell them that you were interested in going to city x (a few hours away) and would like to know if there are any obligations you have to meet on those two days. Or tell them you are meeting a friend.
If they say there are no meetings, but then they fill those two days with meetings they have given a big sign about how they view work/life balance.