Why do people say American cars are now as good as Japanese and South Korean cars?

Many people said that American cars became as good as Japanese cars in 2000.
However, the car my dad used to have, a 2000 Chevrolet Impala, had the check engine light come on it before it had reached 100,000 miles and it started having more and more problems until we had to sell the car in 2012. My mom's 2005 Mercury Grand Marquis has things falling off of it and makes cheap-sounding squeaking noises when it's being driven as well as when you shut the doors. On the other hand, my dad's 2013 Hyundai Sonata doesn't make any cheap sounds and hasn't given my dad a single problem. The most amount of miles anyone in my family has put on an American car is that my uncle managed to put 160,000 miles on his 2005 Jeep Liberty. But, the car was literally dying at the time he sold it.
Horrible experiences with American cars is what has gotten my family to leave Ford and GM for Hyundai (and Honda too, as my mom is thinking about replacing her Mercury with an Acura).
In case you were wondering, my family has always done proper maintenance on their cars.

Any car made by any company can have all sorts of problems. I have seen Toyota's, Honda's, BMW's, & Kia's with all sorts of problems. Never seen a manufacture that has not built a car with some problems. They are mechanical devices built by man so they are not perfect. Have never been and never will be. Overall the Japanese still have a slight edge in reliability. But, American manufactures have closed the gap so much over the past decade that it is really not an issue. Almost any 2015 car you buy will be dependable. Of course there are going to be those cars than have problems. It is inevitable.

Not all cars are equal when they roll off the assembly line. American cars have a long way to go. Though all cars have their own issues, American ones have earned themselves a bad name since the 70s for how more frequently their cars have problems and the Big three haven't gone about cleaning up their image. US automakers make the same mistakes over and over again. Where Japanese automakers and Hyundai differ is they smarten up. Japanese cars in the 70s were outright horrible. However by the 80s, they corrected their mistakes. With Hyundai, they messed up big time in the 80s and by the end of the 90s, they restored their credibility. As for your question, you are just repeating a headline. The media these days has zero credibility, so carefully discern what is true and what isn't. For the Grand Marquis, squeaks and rattles develop over long periods of time. And for the Liberty, even if it were a Japanese car that old, you'll still have some problems.

Because how the mentality of the worker also contributes to the build quality. A Toyota made in America will have more problems than a Toyota built in Japan.

What you've heard is not quite true. Although American car brands have come a long way, so have the Asian and European brands.

Consumer Reports in their annual reliability survey shows Japanese and European brands in the top 11 spots, Buick, GMC, and Chrysler are next. Out of 23 different brands, the bottom 5 brands are American -- Chevrolet, Cadillac, Dodge, Ford, and Jeep.

You are comparing a 2013 car to a 2000/2005 model which is not fair. The newer 2013 American vehicles compare favorably with the Sonata.