So ever since I got Catalina my webloc files have been showing generic icons. Assuming that’s not a Catalina feature, I resorted to the old adage of rebuilding Launch Services. But no change at all. Anybody else see something similar? Any other ideas?
No ideas, I’m afraid, but I’m not seeing the problem here in Catalina. .webloc files have the proper Safari-like icon.
Which reminded me to check, but yes, the default app for webloc files is listed as Safari too so… 
A few more suggestions here:
But if it’s only .webloc files, perhaps try assigning them to another app and then back.
The essence of that page is rebuild LS. Didn’t work.
And I cannot actually set another app. It’s all grayed out in the Information panel.
I have a related issue with Catalina concerning the Time Machine disk icon:
On my MacBookPro, when I plug in the disk, the normal greenish icon shows up to represent the disk. However, on my iMac, I see a generic orange external disk icon. Through the Time Machine System Preference, I can re-designate the disk as the Time Machine disk and the normal Time Machine icon now represents it. Note that the system recognizes it as an existing Time Machine disk and updates it incrementally. Also note that if I do nothing, Time Machine processing occurs normally for the disk, but the icon remains generic.
I’ve seen that with TM too. But mostly in HS and Mojave actually.