The key is layer blend modes. SVG doesn't support layer blend modes (or layers), so Inkscape engages in some trickery to make it work. That trickery includes applying SVG filters to objects in the layer.
If you have at least two layers, and your source layer has a blend mode, when you paste, the new object will have the hidden filter.
Did you intend to use a blend mode? It's possible to accidentally change them, perhaps while using the mouse wheel or a trackpad's scrolling behavior while pointing at the blend mode option. You'll find the blend mode in the Layers dialog. If you don't want blend mode behavior, change it to "Normal." If the Layers dialog is not visible, it can be opened with Layers > Layers... from the menu, or Ctrl+Shift+L.
The immediate solution is to Filters > Remove Filters while the object is selected. If you're using blend modes, you may have to learn to live with it.
(Real credit for the answer goes to Lazur and Moini, who saved me after too many hours of tearing my hair out after unintentionally changing the blend mode.)