I've got a slightly different behavior. For some reason javahome will always return "/Library/Internet Plug-Ins/JavaAppletPlugin.plugin/Contents/Home" regardless of what I give it with the -v flag.
If I say "javahome -V", it will show all the VMs...
$ /usr/libexec/java_home -V
Matching Java Virtual Machines (5):
		15 (x86_64) "AdoptOpenJDK" - "AdoptOpenJDK 15" /Library/Java/JavaVirtualMachines/adoptopenjdk-15.jdk/Contents/Home
		13.0.1 (x86_64) "Oracle Corporation" - "Java SE 13.0.1" /Library/Java/JavaVirtualMachines/jdk-13.0.1.jdk/Contents/Home
		11.0.5 (x86_64) "Oracle Corporation" - "Java SE 11.0.5" /Library/Java/JavaVirtualMachines/jdk-11.0.5.jdk/Contents/Home
		1.8.271.09 (x86_64) "Oracle Corporation" - "Java" /Library/Internet Plug-Ins/JavaAppletPlugin.plugin/Contents/Home
		1.8.0_271 (x86_64) "Oracle Corporation" - "Java SE 8" /Library/Java/JavaVirtualMachines/jdk1.8.0_271.jdk/Contents/Home
But again, no matter what I try to do to get a real JDK, it always gives the same /Library/Internat Plug-ins directory, which isn't a full JDK, so all development apps break.