Should the building inquiry with exemption be rejected?

The house with the two-story extension is therefore also located in an area that provides for single-story construction.

Just because an extension appears two-story, is very tall, or actually is a two-story extension, the house including the extension can still be single-story. The whole counts after all.
Here the rule of single-story construction also applies. At first glance, it consists of one third two-story urban villas and Bauhaus-style houses, which are, however, mathematically assigned to single-story: here a bay window, there a ground-floor flat roof, air space always works as well.

Exemption according to § 31 para. 2 Building Code

I see this more in terms of social compatibility in cases of housing shortages and local space shortages. For example, container construction on public land or land use due to public interest. (Asylum accommodations and expansion modules at schools would be examples.)
Or if a multi-family house gets another floor added to create more apartments, even though there are no additional parking spaces. Then the common good through the creation of living space prevails. But I am not an expert, I can only read.

It may also be justified if the floor area ratio/floor space index for a single-family house for one family would not be sufficient, because the house would then only be allowed to be built too small. For example, if you have a 700 sqm plot with a floor area ratio of 0.1.

We have therefore referred to the granted exemption and therefore do not see ourselves as a "role model" but rather as "imitators."

I hope that is written nowhere, because one definitely does not want imitators.

How big is their extension? and was it definitely built legally?

The extension is about 25 sqm and was built legally. The original building is about 90 sqm.

Then the cultivation is still below the maximum limit and unproblematic?

Just because an extension appears to be two-storey, is very tall, or is indeed a two-storey extension, the house including the extension can still be single-storey. What counts is the whole.
In our case, single-storey also applies. Visually, it consists of one third two-storey city villas and Bauhaus-style houses, which are, however, assigned to single-storey in terms of calculation: here a bay window, there a ground-floor flat roof, open spaces always possible.

I see this rather in terms of social compatibility in the event of a housing shortage and local lack of space. For example, container buildings on public land or land use due to public interest. (Asylum accommodations and extension modules at schools would be such an example)
Or if an apartment building receives an additional storey to create more apartments, even though there are no additional parking spaces. Then the public good of providing housing prevails. But I am not an expert, I can only read.

It may also be justified if the floor area ratio/storey area ratio for a single-family house for one family would not be sufficient, as the house would then have to be built too small. So, if you have, for example, a 700 sqm plot with a floor area ratio of 0.1.

I hope that is not stated anywhere, because you simply don’t want imitators.

Yes, I know the "tricks" with which you can achieve single-storey. We had originally hoped ourselves to get something out of it in this way. However, in our case the building regulations of the 1960s still apply, as the development plan dates from that time. Back then, one third of the floor area had to be below 2.30 m. How large the lower storey is does not matter at all.
In our case, the neighbouring house has a knee wall height of 1.60 m. Due to the then common steep roof pitch, the clear room height of 2.30 m is naturally reached quickly. The extension eliminates the roof pitch over a large part of the rear of the house. I really don’t know where there should still be one third below the relevant clear room height.

In my opinion, other builders are also allowed to invoke a once granted exemption.

For us, it is annoying that we have extensively justified the requested exemption in writing as required and that the hearing now does not address this at all. For example, it would be helpful for the hearing if they had explained to us why the neighbour’s case should be different from ours. Now we have no idea what else we should write except what we have already written anyway. I actually know hearings to be different.

Then the extension is below the maximum limit and unproblematic?

How do you derive that now from the two values?

Back then, 1/3 of the floor area had to be below 2.30 m. How large the lower floor is, unfortunately doesn't matter at all.

the 1/3 refers to the area

In our case, the neighboring house has a knee wall of 1.60 m. Due to the then common steep roof pitch, the clear room height of 2.30 m is of course quickly reached. Due to the extension, the roof pitch is omitted over a large part of the rear side of the house. I really wouldn’t know where there should still be 1/3 below the relevant clear room height.

So, from a distance you can only guess. We basically know nothing except your statements. But these can be subjective. Certainly, there are also facts missing for us that are not mentioned now but which an authority would use for examination. In this case, it might have worked because only the rear side is affected, but the external character remains.

What is annoying for us is that we have extensively justified the requested exemption in writing as required, and this is now not addressed at all in the hearing.

As already said, we do not know the justifications.
And those that could refer to §31 are not that easy to explain in private...