I landed on your question because I was trying to join symmetrical objects but had forgotten the details and was trying to recall how to do so again. While the answers already given here are technically correct, I think there are a couple quirks about how Inkscape works and describes things that are making it confusing.
First, the documentation - while having examples calling out "end nodes" vs. "nodes" - does actually make it seem like any two nodes can be joined.
Join (merge) nodes - Select at least two nodes. When you click on the button, the nodes will be merged into a single node. Inkscape will try to preserve the path’s shape as well as possible.
I believe the observations of others here about this being end nodes only is correct and needs further consideration; however, more on that later.
This can make for a frustrating experience. For example, consider making something symmetrical. You might draw half, copy, flip, and then wish to join the two halves together.
Suppose I have these two halves of a W-shape and then wish to do a "Join nodes" operation:
While the documentation makes it seem this should somehow work, it does not.
However, let's return to the idea that perhaps these need to be end nodes instead. When I think of end nodes I typically think of an open, non-filled path. Joining works fine for a simple case like this:
But all that's really needed is that the path be open - so interestingly we can transform our nodes into end nodes. If we go back to before we copied the object, we can select the segment attached to the node we are going to join and delete it using "Delete segment between two non-end nodes".
Confusingly, the end outcome of this operations looks identical visually because our stroke is not particularly obvious. However, now the operation from before works as expected.
And you can also do the same to the other node affected by deleting the segment since it is an end node now, too.
Now what about the documentation not indicating that these need to be end nodes? Well, I think there's a reason for that, too. It doesn't strictly have to be end nodes if you are also considering nodes in the same path - then it simply appears that they need to be contiguous. For example, you can do this type of "Join nodes" operation:
Which results in:
However, non-contiguous nodes will not work for the same reasons others have listed that arbitrary nodes in different paths cannot be joined. For example, the following will not work:
So returning to your original shape.
We can still get close to a bowtie. First, use the "Insert new node" option on the top segments, like this:
Reposition so that the newly added nodes are similar to the trick we used from the W above.
Now delete the segments to make them end nodes and then join as with the W:
So it isn't strictly one point, but with a bit of positioning most of the same properties are preserved - and most importantly, the shapes are joined.