Can the arduino be recognized if the usb chip is broken?
It depends on the exact arduino product. An Arduino Uno has a USB programmer chip, that then programs the AtMega 328 Chip. If the programmer is broken, it will likely not register any device when you plug in an arduino uno to your PC. 

if the micro-controller is dead, but the programmer is operational, you will get a device in your device manager, but not be able to reprogram the micro controller. 

If both the programmer, and the micro-controller are broken, then figuting that out could be difficult. The AVR microcontrolers support other programing methods, and the AtMega 328 IC, supports DebugWire. you could attempt to reprogram the MCU out side of the programmer, if you thought the programmer, was defective. 

Here is a great article on using DeBugWire.
https://awtfy.com/2010/02/21/modify-an-arduino-for-debugwire/

another nifty trick I learned, is to take your Digital MultiMeter, set it to diode test, or continuity. then, measure between the ground pin, and the positive voltage pins on the MCU.  If you get a shot, the chip is busted. You will expect to get roughly 0.7Volts, although an OL reading might indicate more about your multi-meter, than the MCU. 

Other Arduino devices, that have integrated USB programing, may behave differently. Features of the specific MCU dictate how it can be debugged. You might want to read the datasheet for a specific MCU if you're having trouble.