Does Cygwin have a command equivalent to
sudo mount -o loop factoryfs.ext4 factoryfs? If so, what is it?
No, you can't mount filesystem images using Cygwin. Cygwin mount supports only mounting Windows directories/disks to directories inside Cygwin. See for example http://www.cygwin.com/cygwin-ug-net/using.html#mount-table
Workaround for some filesystems is to mount disk as normal Windows disk and then mount that to Cygwin. For example ext2 is supported in Windows with additional drivers.