Apple has the target disk mode, but doesn’t the laptop need to be shut down for it to work?

Modern Macs can’t do Target Disk Mode. If you had the right cables (thunderbolt or firewire) it was really fast, just as quick as a high end internal PCIe SSD.

And yes, you did need to reboot - because the other computer had full arbitrary read/write access to the raw sectors on the drive with no safety checks or security. If you did that while the computer was running normally, you’d corrupt the data on the disk as soon as they both tried to do a write operation at the same time — and also TDM needed to be used with caution - the other computer could easily install a rootkit or steal all your saved passwords.

It’s been replaced with “Mac Sharing Mode” which operates while the Mac is running normally, does have all the necessary algorithms in place to avoid corrupting the disk, full security to authenticate each read/write operation and block attempts to mess with system files, and therefore is orders of magnitude slower than TDM.