7-seg. Display Multiplexing Disturbing Real-Time Clock
As your colleague cannot provide the schematics or PCB layout, any answer can only be a "best efforts guess" as we can't see what the design actually is.


  using a separate PCB for the RTC block or using 7-seg display modules solves all the problems


This tells us that moving the RTC, or the different schematic &amp; PCB design for the 7-segment displays, is enough to solve one or more of the problems with the design - at least enough for it to appear to work (although some design problems might still remain, even when a design appears to work).

RTCs typically use a low-power crystal drive signal (to save power) and high impedance input, but this makes them sensitive to interference from nearby signals. If such interference happens, those additional transitions of the OSCI signal will make the clock run faster, which is your symptom.

Moving the RTC to a separate PCB obviously increases the distance from potential interference sources to the OSCI signal, so this suggests one or more interference sources are coupling into that signal (or coupling into the OSCO signal, which then affects OSCI).

NXP provide a helpful document - UM10301 User Manual for NXP Real Time Clocks PCF85x3, PCF85x63, PCA8565, PCF2123, and PCA21125 - which includes design guidelines to avoid this type of problem. Your colleague's existing design will likely be breaking one or more of these guidelines, which is causing the problem you describe.

It is best to read the whole document, but one of the relevant sections is section 14 which I have included below. This includes the PCB layout guidelines. Follow them. It is possible that your colleague's design will start to work correctly before making changes to follow all the guidelines, but all of them should be considered for the best noise immunity and design reliability.