using late 2013 imac as display for 2021 macbook with M1 chip

See this requirements for target display mode

To use any of these iMac models as an external display:

• The iMac used as a display must have macOS High Sierra or earlier installed.
• The other Mac that you're connecting it to must have been introduced in 2019 or earlier and have macOS Catalina or earlier installed.
• The cable connecting the two Mac computers must be a Thunderbolt  or Thunderbolt 2  cable.

As you can see, your new MBP is not compatible with target display mode. As a first, it has difficulties to work on a laptop with any OS higher then Catalina.

As a second, you will need to use a thunderbolt 2 cable and your new MBP does not have thunderbolt 2 ports. Target display mode will not work with dongles

Use your iMac as a display with target display mode - Apple Support

Then you should post to the actual forum area for MacBook Pros instead of the one for 12" Retina MacBooks models 2015 to 2017.

It helps to post to the correct forum area. You're more likely to get help there.

The last MacBook was the 2017 model and it didn't have a M1 chip.