Trying to use my old iMac 27" (late 2013) as a second display. Is this compatible?

No. The last Macs that could use a TDM-supported iMac as a monitor were the 2019 Intel models running macOS 10.15 Catalina or older. Your Macbook Pro is far too new. See the very narrow system requirements here:

Use your iMac as a display with target display mode

Newer models including the M1 Mac cannot do it.

TDM is dead technology. Apple started its deprecation in Late 2014 and threw the last shovel of dirt in its face in early 2020.

A free-standing monitor is far superior:

• Cheaper cabling--the iMac may need up to US$100 in Thunderbolt cabling and adaptors.
• Easier to set up
• More reliable
• Supports extended desktop--TDM could only mirror
• Uses less power

lruiz2106 wrote:

I have a new 16" Macbook Pro with M1 Pro chip (2021) and I'm trying to use my old iMac 27" (late 2013) as a second display. Is this double? I have tried the Thunderbolt 3 to TB 2 adapter with the thunderbolt cable. I've also tried using the USB-C to mini display port. None seem to work.

hardly seem possible—

Transfer files between two Mac computers using target disk ...

Use your iMac as a display with Target Display Mode - Apple Support

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

see—

Use external monitors with your Mac: Connect a display to your Mac - Apple Support