I made a circle and 2 short paths (see attachment "Circle_and_2_paths.png"). The Fill is empty, the stroke has a width of 1 mm.

I select the 3 items, and then use "Path/Union". What happens is the the circle and the path on the left combine, but the path on the right disappears (see attachment "One_Disappears.png"). I have no clue why!

I removed the short paths, and draw them again (draw freehand), even in different order (first left, then right or first right, then left), but it's always the right one that disappears when using "Path/Union".

I'm using the latest version of Inkscape (1.0.2-2) on a Windows 10 Home edition.

Happens also to me, anyway using pen tool instead of freehand could be a good workaround (at least, to me).

Edit: using freehand, you'd better convert all from stroke to path before union.

I think it's more of a mystery why it works for the original 'ear' at all. 

If you switch to the node tool ( N or F2 ) and click drag the path of the 'ear', you will find it is in fact converted to a loop !

This makes sense as it it is a thin line and Inkscape tries to make a continuous path by going up and down the 'ear'

Perhaps Path>Combine is what you want instead.

As Mr Riva pointed out, you can do path to stroke, that means you are effectively adding a rectangle to a circle.