There must be something else going on here. The version of VMWare Tools shipped for Windows XP (file /Applications/VMware\ Fusion/Contents/Library/isoimages/x86_64/winPreVista.iso) hasn't been changed since 2016. It identifies itself with VMWare Tools version 10.0.12 which was the last version that was built for XP.

Some basic questions and thoughts

Have you made sure that the guest operating system is configured with a guest OS type of Windows XP Professional?

Have you tried manually assigning the ISO file above to the CD/DVD drive and trying to install Tools from that?

Perhaps VMware didn't test Windows XP with Fusion 13 because Windows XP is seriously end-of-life from Microsoft. 

Have you tried dropping the virtual machine compatibility to hardware version 18 or 19 and see if that makes a difference?

Could you attach the vmware.log file for the virtual machine where the installation of Tools fails?