Doesn't Arduino reports DSR (Data Set Ready) correctly?
No, it does not, and can not for the simple reason that there is no DSR line connected to the processor.

Generally host software / serial driver configuration ignores DSR.

On the boards with an 8u2/16u2 the custom firmware therein can report whatever virtual state of the signal it is programmed to. On those with hardware USB-serial they will report however the pin is hard wired, or whatever value it is pulled to or floating at.  

None of these (possibly excepting a Leonardo type board or an ARM one with native USB) reflect anything about the state of the target processor, so software hoping to talk to an Arduino should ignore the DSR signal.