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I'm struggling with the merging or joining of two paths.
I have two paths, the red and the black one, when I join them with the join tool it ends up looking like on the second image.
Can anyone explain to me why this is happening and what would be the better way to create such a path?
What you need is not possible with Illustrator's path logic.
Paths need to in be a continuous, one-after-another anchor point sequence.
Instead, you can either group these, or convert to outlines, in which case you can join multiple shapes into a single shape, but you're losing the ability to edit these as paths (unless you keep an un-converted copy).
No branching path is possible. Use grouping.
Or outline and pathfinder for a outlined shape.
But this question opens a good learning opporunity:
The purpose of a vector file is not to make it possible to model whatever you want how you want it. Instead the purpose is to make a minimal set of features that are needed to render a particular feature to rasters. There's never a need to do this, 2 lines works just as well from the rasterization aspect.
Why minimal? Because a vector file is not a at all a file like most image file formats. A vector file is more like a programming language. And every program that renders the image needs to run the vector file. So implementing extra features is a problem, it costs money to do and maintain.* Having super high level features is also in practice a problem for users. This extra complexity increase should be taken only when necessary not for personal convenience.
Therefore, it is your, or a higher order software's, job to convert whatever the internal representation you or software is using into this format. Illustrator just happens to be a direct modeler so its quite close pedal to the metal.
Incidenttaly this also limits features vector drawing software can have. You can only do things that you can convert to the minimal description.
* This is one of the reasons why SVG has taken so long to adopt and even then only part of the original design is adopted widely.
Joincommand does not mean "merge paths".Joinmerely connects the two closest end points of separate paths to one another (i.e. your second image). You'll find, in your second image, the vertical stroke is overlapping itself in the top portion... it's not "merged" into a single path without overlaps.