Note: this relates to dual citizenship by naturalisation (durch Verleihung). Children of parents of two citizenships, of which one is Austrian (e.g. Austrian mother and British father) may now hold dual citizenship by descent (durch Abstammung).
From September 2020 there has been the possibility for (re)acquiring citizenship by declaration (durch Anzeige) for the victims of persecution under the Nazi regime (who fled Austrian up until May 1955) as well as for their direct descendants. The party submitting the declaration is not required to renounce their previous citizenship in this case. The only other case of dual citizenship being allowed through naturalisation is where citizenship is awarded in the interests of the Republic of Austria (Verleihung im Interesse der Bundesrepublik Österreich).
There have been several initiatives to try to allow dual citizenship, predominantly for Austrians wanting to take a second citizenship and maintaining their Austrian citizenship. This is already possible by exercising a right to retain Austrian citizenship (Beibehaltungsrecht), although the justification to do needs to be well substantiated rather than perceived as trivial.
Any such change would need amendments to be made to Austrian law and therefore require significant political backing since the process would have to make its way through parliament, and many Austrian political parties are not known to be in favour of any relaxation of the law.
It is infeasible that such a landmark change to the law would be made for a single country, and political sensibilities might be particularly averse to any relaxation for citizens from third countries.