Multiple Macs - Multiple Time Machine - multiple(s) Drive issue

[smogsy]

I got possibly a rare issue but wonder if there is a way to fix it

Issue
when switching from Mac Home to Mac Work or via versa The other laptops Time Machine wants to connect to the laptop & spams do you want to connect X drive if I hit cancel it just reappears.
only workaround found is use another Thunderbolt port to connect to each mac

What I want
everything connected to the caldigit dock & a way to ignore that drive

Setup
Home
MacBook Pro M3

work
MacBook Pro M1

both are connected to Caldigit TS4 element

 

[smogsy]

I tried it again after I setup my home Mac with TM & second drive ask once & seem not to annoy now.. maybe its fixed?