""... the only difference of asynchronous and synchronous data transfer is the presence of clock signal ..."

This is true.
Note that RS232 covers both synchronous & async.

"How do we implement asynchronous data transfer (without bit-banging) on standard-8051 device that only have UART port?"

It's got a UART - use it!
If you mean you want more than one async transfer but have only 1 UART, then you will either have to do it in software ("bit banging") or use external hardware.