You are considering claiming damages due to the depreciation you feel caused by the non-compliance with building regulations. Any claim on our part is based on claims. These can be found here in the Building Code. The identification will presumably only be able to rely on three legal grounds. §§ 280 and 823 I Building Code are excluded, since no obligation exists towards the neighbor under 280, and under 823 I financial damages are not covered.
Thus, only § 1004 Building Code remains. § 1004 I Building Code provides a claim for removal, which is limited in 1004 II in that it does not apply if you are obliged to tolerate it. Furthermore, it has emerged that the removal must also be reasonable, meaning it does not have to be carried out at any cost.
How does it stand here? The rascal simply (?) built too high and the municipality noticed and imposed a fine. At the same time, it did not order the dismantling, presumably due to the disproportion.
Of course, you can have any claims reviewed by a lawyer of your choice and try to enforce them. I do not want to spoil your fun, but I fear that this action will be as expensive as it is unsuccessful. In addition, the relationship with the neighbor (and possibly other neighbors) would be ruined forever, I would assume.
Swallow your anger and enjoy your beautiful home. The impairments really remain within limits, and a real financial disadvantage would only arise, if at all, when you sell the property.