Welcome to the community Silas!
It all depends on how you exported the license. From the LTU How-To:
Private Export – This is the default method and restricts the license import to your Autodesk ID to ensure it is not transferred to another person. Private exports automatically become public if not imported within 14 days.
Public Export – This method means your license can be transferred to someone else running the same product and serial number, such as another user in the same company. To use this method, check “Make the license available as a public import”.
Please accept as a solution if this fully answers your inquiry, or reply with additional specifics as needed.
You're welcome! The other user would simply need to enter the same serial number during the installation process and then use their Autodesk ID / password to access the license via the LTU.
If nothing is showing as available, then chances are the license was privately exported (default) in which case you would want to log into the LTU on the users computer to access the license.
No worries / keep them coming
The computer name should not have an impact on utilizing the LTU as the system is only looking at the current status (can't export if the SN is already exported from a different machine), serial number (needs to match up with the one used on the install) and the user.