Unexpected Seven Segment Output

I won't throw any code in but I'll make these observations:

You have two LED modules. Each LED module can display two seven segment digits. Each module has its own SN74HC595. The shift registers are daisy chained. Is that all correct ?

Your numbering of the digits (Digit_1 etc.) seems odd because of way Hours and Minutes are mixed.

  pinMode(Digit_1, OUTPUT); // Mins Tens
  pinMode(Digit_2, OUTPUT); // Hrs Tens
  pinMode(Digit_3, OUTPUT); // Hrs Ones
  pinMode(Digit_4, OUTPUT); // Min Ones

It is difficult to imagine how these two displays are actually viewed. Is one above the other ?

As @ShermanP pointed out, your multiplexing is odd (if indeed you do have two shift registers).
Normally, and to maximise brightness, you'd simultaneously display 1 digit on each display. This means pushing out two digits at a time to the shift register chain.
You also save 2 arduino pins because you can join the cathodes of corresponding digits (assuming the load (mA) isn't too much, that is)

As it is, you appear to be displaying only one digit at any one time. Further, the effect you are noticing of mixed up digits is probably because, at the time you are switching on a particular digit, the matching data is not shift register belonging to that digit.

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