External SSHD has Mac OS X installed by another mac. Can I use that SSHD on my Mac? There was an external SSHD that had MAC OS X installed on it through Internet Recovery by using another Macintosh with a USB 3.0 to Sata II cable. Is there some sort of serial number that is tied to that installation of Mac OS X on the external SSHD? Or can I actually boot that external SSHD on another Macintosh and it would be fine?
There is no serial number (like a Windows 7/8/10 product key) tied to the OS X installation. If you didn't configure iCloud/iTunes accounts while setting up the system it's almost plain vanilla.

But installing OS X in Internet Recovery Mode will install the system the Mac originally was shipped with. Moving the disk with this system to a newer Mac - which usually relies on a newer OS X version - will probably fail to boot.

Usually newer Mac contain newer or additional hardware. The older system usually doesn't ship with the newer or additional drivers needed.

Example:

The iMac Late 2011 (Mac12,1) was shipped with OS X 10.7.2. The iMac Late 2013 (Mac14,2) was shipped with OS X 10.8.4. Attaching the external disk to the iMac Late 2011 and using IRM's Restore OS X will install 10.7.x on it. The same disk attached to the iMac Late 2013 will probably fail to boot because the system is too old.