I have never worked with 74hc595, and I wonder if I could drive an common anode 7-segment with it or no!? because I think 74hc595 is not open collector on output and anode 7-seg types needs to drive a current to ground to turn a segment on.
It would work for a single digit display per 595, as long as the anode is using the same power supply as the IC and you use individual segment resistors to limit the current to less than 8mA per segment.
If you need higher current than that, I'd suggest using a different driver such as an MIC5841; that has a serial input & latch similar to a 595, but can sink 100mA or more per output.
Unless you only want to drive ONE set of seven segments, it's not a great idea; open collector
drivers and time-division to drive multiple digits means the segment drive has to be high current
because the common anode is switched OFF most of the time (while the other
digits are driven). ULN2003 can sink a few hundred milliamps per segment, but the 74HC595
can only sink 20 mA safely (at circa 5V).