These cases have been well covered by German media as well, and since both the tenants, the location and the municipal politicians have been named, the cases can easily be verified and I don't see any reason to doubt it. Just to mention a few of the articles about this specific case:
What is not so clear from the Daily Mail article is that Gabrielle Keller is renting an apartment from the municipality, a rather uncommon situation in Germany, since the municipalities themselves usually don't own houses usable as living space.
The termination of (housing) rental contracts is regulated in the German Civil Code (Bürgerliches Gesetzbuch) §573 which rather vaguely states:
Der Vermieter kann nur kündigen, wenn er ein berechtigtes Interesse an
der Beendigung des Mietverhältnisses hat.
My English translation:
The landlord can only then terminate (the contract), when he has a
justified interest in the termination of the tenancy.
The civil code continues with an incomplete list of justified interests. Most commonly, the termination of a German rental contract is justified with the landlord's personal need.
The highest court in Germany (Federal Court of Justice of Germany, Bundesgerichtshof) has in their verdict VIII ZR 238/11 already ruled that the 'justified interest' requirement is fulfilled for a legal entity (in this case, the municipailty as landlord is a legal entity and not a natural person) if the entity needs the housing space to fulfil official duties.
Even if the exact circumstances are different in the high court verdict, the German municipalities have been required to shelter a given number of refugees and they are hence justifying the cancellation of the rental contracts with their demand for housing space to fulfil their official duties. In most of the covered cases, the current tenants have disputed the termination notice and if it comes to a court ruling, the courts may of course rule differently, since the surrounding circumstances are different.
Also very different opinions have been expressed on the legal standing of the termination of the rental contracts. In the article from the Badische Zeitung, Udo Kasper from 'Mieterbund Baden-Württemberg' (a tenants' association) claims the termination to be void, since equal interests and requirements of two tenants are played against each other. He continues: 'The last consequence of the situation would be: The municipality evicts the current tenant. The municipality would then be responsible for finding accomodation for the now homeless woman. They would have to provide her with living quarters.' So far so good, however the municipality would not be required to provide the current tenant with a 78m² appartment, in which she lives alone.
In a followup article in Focus, 'lawyers' (they are not named) consider the termination void, since it is justified with the municipality's own need. According to the quoted lawyers, the concept of 'own need' or 'personal need' (Eigenbedarf) as defined in the German Civil Code can allegedly only be applied by natural persons and not a municipality. This opinion is however in contradiction with the court verdict I linked to, in which the highest court has already ruled that the principle of 'Eigenbedarf' can be claimed by legal entities as well.
In the Die Welt article, Wiebke Werner from another tenants' association finds the termination not necessarily disqualified and states: 'it depends on the assessment of each single case'.