I'm very new to Inkscape and I'm sorry if I'm using the wrong terminology.

I would like to understand how can I draw a line using Bezier that starts from an existing line (see image) and this new line belongs to the same path of the preexisting line. So, since they belong to the same path if I move one line, also the other one move like if they are the same object.

I tried to do Path --> Union, but for some reason one line disappears.

Thank you

Path > union must be meant as union of areas (shapes, fills). It does not work on lines (strokes).

Instead use path > combine.

To answer your question, one node can't be linked to more than two nodes. There are no "T" nodes (linked to 3) but ypu can stack one node connected to 2 and one connected to 1 to get the same visual result.

Turn on snapping to nodes > cusp nodes and smooth nodes. This way, you can move one end node on another one. Then select both paths and do path > combine (this step can be skipped) and rectangle select the two nodes : then you can move them in one go.

"Linked" is perhaps the wrong term; you can´t "join" more than 2 nodes but move e.g. many nodes at once: