Disadvantage of a building encumbrance in our specific case?

Hello

ok, if that is the case, the OP has now received enough input.
And I learned something.

Steven

As far as I know, it is the case that a different subsequent neighbor would no longer have to tolerate this garage, even if you come to an agreement with the current neighbor – but I am not an expert on this. Our garage partially stands on the neighbor’s property, and for that they were given a piece of our property to use back then; this agreement was made between the previous owner of our house and the current neighbors. We took it over and left it as it was, but we could have also dissolved the agreement. However, if the owners over there were to change, they could insist on reversing this agreement. Then we would have to dismantle the garage and they would have to give us back our garden piece. There is no construction burden registered in our case, only a note about this matter was included in the purchase contract.
The situation is similar with trees that are unlawfully large and stand too close to the fence. Here too, a new neighbor can demand that the tree be removed. For example, our neighbors have a huge thuja (as tall as our house, and we have two upper floors), which stands almost directly on the boundary. We could have demanded its felling within a certain period (we did not do so and do not intend to).
I think this is a factor not to be underestimated, for which one should seek advice. If that really applies, then in your place, I would simply build the garage correctly from the start and thus avoid possible trouble in the future.

I am not an expert on this

Then how do you come to your assertions?

that a different subsequent neighbor would then no longer have to tolerate this garage

What does this statement refer to? What does the "then" refer to? What should the neighbor be able to do?

but we could also have terminated the agreement.

If the agreement should be in writing – after all there is a note in the purchase contract – or if there were witnesses, that would not be easily possible.

Then we would have to dismantle the garage

That is very unlikely. Claims for dismantling usually have to be asserted very promptly.

It is similar with trees that are unlawfully large

A tree cannot be unlawfully large.

Here, too, a new neighbor can demand that the tree be removed.

No, at least not in NRW. Which federal state are you talking about and on what legal basis should this right be founded?

I think that is a factor not to be underestimated, for which one should seek advice. If that is really the case ...

What are you talking about? Which factor do you mean? What should apply to what?

I am not an expert, I have pointed that out. I am simply repeating what was told to me when we bought our house. Besides, it would be news to me that you are only allowed to respond here if you are an expert. Anyone who wants legally binding advice must go to a lawyer or similar.

But please excuse my existence.

By the way, it would be new to me that you are only allowed to reply here if you are an expert.

You didn't reply at all. You didn't address any of the OP's questions. Instead, your post raises many questions that you also do not answer. By the way, I haven't forbidden you from replying, but merely tried to find out what you wanted to communicate to the OP or the forum.

Honestly, I also don't understand why OPs are often unsettled by multiple subjective opinions, hearsay, half-knowledge, or the like although the designated professional/expert has presented everything coherently and, above all, reliably. is a surveying engineer; it's simply brilliant that one can receive questions from him so concretely and thoroughly here without much effort, and for free, always backed up with sources and paragraphs...