So I'm creating a logo with the multiply blend effect on one layer, and there's a part I made on that multiply/layer that I copied and pasted to one above it so I could make it opaque. But the multiply effect is still part of the object I copied despite it now being on a normal layer. How do I remove that effect so that I can make the part opaque?
Select 'Filters' > 'Remove Filters...' - the blend mode is just a filter applied to a layer. Objects in the layer inherit the filter from the group (=layer) they are in.
If you copy one of them out of the layer, it keeps the filter.
It's the same behaviour for groups (and layers are just groups in Inkscape).
So, it's maybe unexpected if one doesn't know how Inkscape layers and blend modes work, but very explicable.
You have found some obscure behaviour. Here's a green rectangle in the bottom layer and a blue rectangle in the mid layer which has layer blending mode = multiply.
When the blue rectangle is copied and pasted to the top layer, the copy has blending mode=multiply. It's bounding box seems quite big. The reason=Unknown.
If the blending mode of the copy is changed to Normal, the multiply layer also turns to normal and the original blue rectangle becomes invisible;
The original blue rectangle still exists. It's selectable in the Objects panel:
The scene is fixed when the layer blending mode normal is turned back to multiply:
I do not know, if this is a known bug.