Yes, but the article also says that Hitler revoked his nationality and her mother’s. So, he did not lose it after the fall of the Empire, as you posted.
In 1949, he (Otto) ennobled several people, granting them Austrian noble titles, although not recognized by the Austrian republic. As he did not possess a passport and was effectively stateless , he was given a passport of the Principality of Monaco, thanks to the intervention of Charles de Gaulle in 1946. As a Knight of Malta, the order also issued him a diplomatic passport. Later, he was also issued with a Spanish diplomatic passport.
On 8 May 1956, Otto was recognized as an Austrian citizen by the provincial government of Lower Austria. The Austrian Interior Ministry approved this declaration of citizenship, but on the condition that he accept the name Dr. Otto Habsburg-Lothringen, on 8 February 1957. However, this only entitled him to a passport "valid in every country but Austria". Otto had already submitted a written statement, on 21 February 1958, that he and his family would renounce all privileges to which a member of the House of Habsburg was formerly entitled, but this first declaration did not satisfy the requirements of the Habsburg Law, which stated that Otto and other descendants of Charles could only return to Austria if they renounced all royal claims and accepted the status of private citizens. He officially declared his loyalty to the Republic of Austria on 5 June 1961, but this statement was ruled insufficient as well.
In a declaration dated 31 May 1961, Otto renounced all claims to the Austrian throne and proclaimed himself "a loyal citizen of the republic", "for purely practical reasons". In a 2007 interview on the occasion of his approaching 95th birthday, Otto stated:
This was such an infamy, I'd rather never have signed it. They demanded that I abstain from politics. I would not have dreamed of complying. Once you have tasted the opium of politics, you never get rid of it.
Archduke Otto and his brothers had an Austrian passport, but they could not enter the country if they did not renounce to their rights. Otto renounced in 1961 and was allowed to visit Austria.
Nationality can be lost too.
Otto von Habsburg lost his austrian nationality after the fall of the Empire.
And the right to live in the country they once ruled in most cases.
Basically everything that is state owned or crown owned and yes in several examples monarchs and members of their family were murdered.
In the UK HM would loose all her palaces except Sandringham and Balmoral as they are privately owned. Most of the collections she has are crown property so would also fall to the nation but everything she owns privately would remain in her possession.
In the Netherlands the King would loose the use of the 3 palaces that are state owned and the use of Castle het Oude Loo but he would get the part of the domaine surrounding the castle that Queen Wilhelmina gave to the state back as that gift is under conditions and in case the monarchy is ended the property would return to the family. Everything that is privately owned by members of the Royal Family and by the various foundations would remain in private possession also.
As much of the collections used by the Belgian monarchy are part of the gift of Leopold II to the state im not certain if he put in stipulations about what would happen to his gift in case Belgium abolished the monarchy.
In most European monarchies there is a clear distinction between private and public (state or crown) owned properties and collections. I think the most tricky countries would be Liechtenstein and Monaco not surprisingly they are also the two richest families.
When a country's electorate vote to abolish it's monarchy, what does the monarch then loose other than his title?
In olden days usually their head. As in the case of King Charles I of Britain and King Louis XVI of France