Bank demands an appraiser, however financing is approved

Now the bank has gotten back in touch and claims that the legislator requires the bank to demand such an appraisal

I am not aware of any legal requirements in this regard. However, I cannot know everything, even though my job gives me insight into various areas.
It might possibly have something to do with the regulations under Basel II.

Perhaps @Voki can contribute something regarding the legal perspective.

If the appraisal report is (always and exclusively) made available to the bank, the bank may not pass the costs on to you. However, if the bank hands over the appraisal report to you after it has been created (or at the end of the loan term), it is allowed to do so, provided there are corresponding contractual agreements.

The appraisal report will never be exclusively for the bank. The bank makes a copy and then gives it back to the client (you).

With this statement, the bank does itself no favor. The reason why the costs are to be borne by the bank is precisely because the appraisal was made in the interest of the bank.

With this statement, the bank does itself no favors. The reason why the costs have to be borne by the bank is precisely because the appraisal was carried out in the interest of the bank.

Here I completely agree with you – with this statement the bank does itself no favors (although, as I already wrote, I do not know all the legal basics).
From time to time, advisors fall back on the point "legal requirement" – either out of ignorance (by equating guidelines/regulations with laws), or because they want to "finally settle" the issue (laws are usually immutable conditions that the customer/the bank have to comply with).
There may possibly be a third point: the matter is unpleasant for the advisor himself. As the OP wrote, the advisor inquired in advance whether an appraisal was necessary and passed on the then information ("No") to the customer. Suddenly he finds himself forced (by new regulations or the like) to deviate from the previously made statement and to revise himself (in my opinion also poor communication style within the bank, since this makes their advisors appear bad/incompetent – unless the advisor’s first statement was already made months ago). So retreating to "laws" is a way out for him.

There is a judgment on valuation fees against a VoBo. It is not a BGH ruling and therefore only applies in this one case and is not a general basis. See: OLG Düsseldorf I-6 U 17/09 Legally necessary and the mentioned wording is of course very general and vague, but the question is how deep you want to get into the topic? The short version is: If the bank is to grant loans against collateral (here real estate), then the value of the property must be known. This is done through the valuation according to relevant legal rules. The triggers are various causes. But that goes too far here. Alternatively, the borrower can also submit their own expert opinion. Then the bank can possibly waive an additional valuation. A valuation can be omitted if the loan is granted without land charges. However, the condition will be adjusted accordingly or the loan will not be approved in the first place.

The report will never be exclusively for the bank. The bank makes a copy and then gives it back to the client (you).

This varies from bank to bank, but in the end you are right... eventually, the owner receives it, and thus the fee is justified (at least according to the judgments known to me so far).