The reason that it's not working is that your M1 MacBook Air does not support USB Target Disk Mode.
Transfer files between two Mac computers using target disk mode - Apple Support
"Note: If either of the computers has macOS 11 or later installed, you must connect the two computers using a Thunderbolt cable."
All Apple Silicon Macs come with macOS 11 or later, so Q.E.D., a USB connection won't work.
A Thunderbolt connection might. However, all of the 2015 Mac laptops (MacBook Airs, MacBook Pros) which had Thunderbolt ports had Thunderbolt 2 ports with the old-style Mini DisplayPort connector. You said your old laptop had a USB-C port.
That pegs it as a 12" Retina MacBook. The 12" Retina MacBooks have USB-C ports – but they do not feature the optional Thunderbolt enhancement. (That makes them unique among Macs, because every other Mac that has USB-C does support Thunderbolt 3 or Thunderbolt 4.)
Your 2015 MacBook can't speak Thunderbolt. For the purposes of Target Disk Mode, your M1 MacBook Pro won't speak anything other than Thunderbolt. Even though it's normally happy to work with external USB drives. There isn't any common ground, and that would be an issue even if the old MacBook's screen wasn't busted.
I looked up a teardown for that 12" Retina MacBook. The flash chip is attached directly to the main logic board; it isn't on any sort of M.2 or similar module that you could easily remove and transfer to an external enclosure.
So that leaves us with a couple of options for trying to salvage your data.
#1: Charge the MacBook. Then plug in an external monitor, turn it on, and hope that you can somehow make use of it with the aid of the working screen. (In a pinch, you could use a HDTV set, attached via a USB-C to HDMI adapter cable. Those cables are easy to find on Amazon, and pretty cheap.). If you don't have a USB-C dock, to allow you to simultaneously attach an external drive, you could still do things like copying files to iCloud Drive, mailing them to yourself, etc.
#2: Forget recovering any data from the MacBook, and try to recover data from your backups (assuming that you have them).
