What are the modern day applications of pc 16550 UART, is it still widely used in the industry?
The UART itself not, but because of existing software, it makes it easier if a uart behaves like a 16550. 

When I was developing a UART for an ASIC, I was asked by my boss to make it 16550 compatible. 

All I had to do was adapt the control registers and map the bits and addresses to the same location as on the 16550. Then move all the others extra features to some registers which do not exists on the 16550.

It worked indeed direct with the standard Linux driver. That saved a lot of time in bringing up Linux as SW people like to use printk for debugging.