Are the Authorities in Austria, allowed to ask for additional documents that they did not previously specify, when one applies for a visa?
Only when the additional documents are relevant to the application being made.
For example, when applying for the post-Brexit EUV 50 card (i.e. "The residency card that all British citizens in Austria, who were resident in Austria prior to 31.12.2020, must apply for in 2021" since they are non-EU citizens), if the required filings (erforderliche Unterlagen) are a valid passport, timestamped photo, correctly filled-out form, and a valid Anmeldebescheinigung and you provide all of these, are they legally allowed to then demand something not mentioned, like your bank details from 5 years ago?
No, since you supplied the Anmeldebescheinigung which proved the date you became a resident.
Had you 'forgotten' to register (as you, as an EU Citizen, were required to do after 3 months), then the bank details could serve as an alternative proof that you were working in Austria 5 years ago. Then the answer would be 'Yes'.
There were also cases where the person registered much later than after the date they actually took up residence.
If they registered in 2020, then they would only be eligible for the temporary card (residence less than 5 years).
If they actually arrived in 2014, they would then be eligible for the permament card if acceptible proof is supplied.
The bank details, showing employment income, (or payments details of contributions into the pension system) could then be demanded to supply the proof needed to issue a permament card. Otherwise only a temporary card would be issued, should you refuse (or cannot) supply the needed documents.
As applicant you are required to supply any needed documents as long as it is relevant to the application.
In Germany, this is called Verhältnismäßigkeitsprinzip (Deutschland) (principle of proportionality), that also exists in Austria and Switzerland.
Since this principle essentially applies everywhere, there is no need to repeat it explicitly in individual laws where it is used.