Since the LM8363 has discrete outputs for HH:MM:SS it drives each segment (20 segments) individually. There is just one grid.
My new VFD needs to be multiplexed, because it just has 7 segments (which are all interconnected which other) but 5 grids.
Is there a way to generate the needed multiplex signal (with for example shift registers?) from the 20 outputs of the LM8363?
Would it be very diffucult or doable?
Somewhere in the middle

That many digits can not be driven non-mux from 20 pins, and data suggests they use a mains half cycle multiplex scheme on the LM8363 for HH:MM:SS
That means you need to sniff those pins, and demux the contents, then re-mux onto your display.
A simple MCU is the best way to manage that, it could just use level shifters, if it has enough pins.
eg If we look for 'modern DIP' (a bit of an oxymoron) I see lcsc have stocks of
AI8051U-34K64-PDIP40, that new part has a USB loader inbuilt, and is in easy to use DIP40, it might just fit with level shifters alone.
STC8H8K64U-45I-SKDIP28 smaller sibling SKDIP28, that could use a VFD serial shift register
You can also buy modules with these MCU fitted from Aliexpress.
AVR mcu also come in DIP.
My new VFD needs to be multiplexed, because it just has 7 segments (which are all interconnected which other) but 5 grids.
No colons or decimal points to flash every second ? 5x7 comes up short for HH:MM:SS ?