No this is not normal behavior.
You need to troubleshoot what is going on, start by getting clear what/when this happens. This happens when you run Pro Tools right? It's not just the action of connecting/mounting a drive.
Pro Tools blindly tries to reinstall files files under ~/Documents/Pro\ Tools every frigging time it starts up. A stupid idea that just slows down Pro Tools start up. What is happening here will likely be an extension of that.
(See
https://duc.avid.com/showthread.php?t=421450 and Case 04496592 ... the underlying "copy everything" behavior here should have been escalated to a bug or attached to an existing bug if one exists.)
Some things you should likely consider:
Have you tried to trash prefs, all troubleshooting starts there, see "help us help you" up the top of every DUC web page.
How exactly are you launching Pro Tools?
Where is Pro Tools app bundle installed? Have you done anything unusual there? It should only be put in /Applications.
Look what is going on with your environment variables HOME and USER, I have no idea how Pro Tools works out the path for the Documents/Pro\ Tools folder but you should be able to work this out if needed.
Is your home directory or Documents or Documents/Pro\ Tools is symbolically linked anywhere?
Is your home directory using iCloud (don't do that).
What filesystem is on the external drive(s)?
If it turns out you are not doing something too crazy to cause this please file a support case and push to get escalated to a bug. The underlying blind copy of all this stuff on Pro Tools start up is just stupid.