How do I connect a 2015 iMac to a iMac M4 for data migration using Thunderbolt or USB?

Get hold of an external drive with at least 2x the storage capacity of the 2015 iMac and create a Time Machine backup.

Then connect that backup drive to the new iMac and migrate from that.

Back up your Mac with Time Machine - Apple Support

Transfer to a new Mac with Migration Assistant - Apple Support

When done, you can continue to use that external drive as a backup destination for Time Machine on the new Mac.

Backups are good, unless your data means nothing to you.

An HDD for Time Machine as was an excellent suggestion, or maybe a NAS if you have multiple Macs and want to share one pool of storage as a NAS. The NAS can be purpose-built commercial with Time Machine server capabilities such as Synology or Unifi UNAS, or can open-source using TrueNAS or ZimaNAS on some spare x86 or Arm hardware and storage, or an existing Mac with directly-connected storage.

If you want to migrate please follow the simple instructions in Setup Assistant and Migration Assistant, this method works 100% of the time assuming 2 things:

1. Your Time Machine backup External Hard Disk is in good working condition.
2. You bought a new Mac with adequate storage space. Adequate storage space means that the new Mac has at least the same or more storage space as the old Mac.