There are a number of posts on these forums where users are reporting issues between M1 Macs and some external drives. Unfortunately the specific details are not clear as only a few users have posted any real details. I know that some JMicron USB chipsets found in some USB drives are not compatible with an M1 Mac which I've seen mentioned on a reputable vendors product page (now deleted) and the couple detailed posts on these forums although other USB chipsets may also be involved. Make sure to disconnect all other external devices when trying to access the old backup drive and connect the external drive directly to the laptop. Maybe even try connecting the backup drive while booting the M1 into Safe Mode.
If you have a 15" MBPro, then you should be able to remove the SSD and install it into an external enclosure. If you have a Retina model laptop 2013-2015 with an original Apple SSD, then you will need to install the Apple SSD into an OWC Envoy Pro Enclosure, otherwise if you have a third party NVMe SSD installed, then it will depend on the make & model of the SSD whether you can use an external enclosure.
If you have a non-Retina model MBPro, then it uses a standard 2.5" drive which you can use a USB to SATA Adapter, drive dock, or enclosure to connect the SSD to another the new Mac.
If the old laptop can power on (does not need to boot or have a working display), then you may be able to use Target Disk Mode to connect the old laptop to the new Mac.
This all assumes the hard drive/SSD is working.
If you are not sure of the exact model of your old laptop, then you can get that information by entering the serial number here:
https://checkcoverage.apple.com/
If the drive is not removable or the drive is bad, then you can contact a professional data recovery service such as Drive Savers or Ontrack. Both vendors provide free estimates and both are recommended by Apple and other OEMs.