Border construction in open building style - Is neighbor approval necessary?

Who asked about the open construction method? It is simply about the garage in boundary development, and this is regulated by the state building code. Exceptions are best discussed with the responsible building authority if the development plan does not provide any information on this.

Look at the thread title

I find nothing about the open construction method in the Lower Saxony Building Code.

Because the construction method is planning law, which is described in the Building Code and in the land use ordinance you correctly referenced.

From this I conclude that nothing needs to be considered for garages.

Your conclusion is correct for planning law. Unless the development plan stipulates otherwise (keywords: buildable plot area, floor area ratio). However, you must also observe building regulatory law, which is regulated in the state building code. If there are no other privileged buildings within the minimum distance to the boundary on your property, you can build your 9m garage without neighbor approval. But if, for example, you already have a garden shed standing on the same boundary or a 6.1m long pigeon loft attached to any boundary, you exceed the maximum values specified in §5 (8) 3 and need neighbor approval.