Building boundary / Fencing - What am I entitled to demand?

With us, the unwritten rule is that everyone takes the right side and sets up a fence.

In our case, the unwritten rule is that everyone takes the right side and erects a fence.

that is probably not unwritten but can be found in the state building code.

I am only concerned with the enclosure and who is obliged to do it

§32 NachbG NRW
If both properties are developed or commercially used, their owners are obliged to build the enclosure together if even one of them demands the enclosure.

there was probably a fence before

After you became the owner?

along the building boundary

You mean the property boundary. The building boundary runs along the street front of your house.

Can I demand that he build a wall, as it serves as a visual barrier??

Yes. However, he is not required to do so, since you are both responsible. Incidentally, §35 NachbG NRW applies:
(1) The enclosure must be customary for the location…

that is presumably not unwritten

Correct.

but can be found in the state building code

Wrong, Brandenburg Neighbor Law (BbgNRG)
In NRW, however, unlike in NI or BB, the right to enclosure does not apply, but rather the joint enclosure obligation.

Thank you first of all for your answers.

I have found the old building permit from 1995 and an old picture.
You can also see the old enclosure there. It is no longer there.
Current state: See picture.

In the end, it is about a clean separation of the areas and unfortunately, that is not the case here.
It can't be that the neighbor simply removed the enclosure and still uses part of my property.
And on top of that, the wooden cladding has become rotten because of it.

In the end, I don't care whether the privacy screen is built from masonry or wood. The fact is that there has to be a privacy screen once I have torn down the terrace.

Preparation in terms of rules knowledge is good for such a conversation. I’m curious how it goes. The entry will be crucial. "You have... and are not allowed to... and must..." is a recipe for making a situation tense. "We have... planned and want to find a solution with you for..." is more promising. They are "only" control instruments within one’s own sphere of influence, no guarantee of success. I’m curious about how the conversation turned out.

Spoke with the owner. Everything is all clear. As soon as I have finalized my plan, he will take care of the enclosure. I will give feedback as soon as it starts.

Regards