In a similar situation, my Time Machine icon became generic when I redid my TM drive. The solution was to do a Get Info on the TM drive. Both the generic and the TM icon appeared in the Get Info window. I dragged the TM icon onto the generic icon and that solved it.
I don't know if this applies to your situation, but it's worth a look.
Jon
macOS 11.7.10, iMac Retina 5K 27-inch, late 2014, 3.5 GHz Intel Core i5, 1 TB fusion drive, 16 GB RAM, Epson SureColor P600, Photoshop CC, Lightroom CC, MS Office 365
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