Thank you for sharing the article. I appreciate that the writer, Hannah Rice, linked to her sources in the French-language press.
The linked 2005 report from L'Express answers the question I asked earlier about whether Prince Albert II consented to his out-of-wedlock children bearing his surname. Apparently he did not, at least as of 2003, when Nicole was still fighting to have Albert's paternity of her son legally recognized.
That means Alexandre is not entitled to change his surname to Grimaldi under either Monegasque or French law, unless his father has since changed his mind.
In Monaco, the Sovereign Family's house law has legal force according to the Constitution (as covered in the "succession issues" thread), and Article 2 states that the inheritance of the name Grimaldi is regulated by Sovereign Decision.
https://legimonaco.mc/tnc/ordonnance/2015/06-02-5.344/
In France, the Civil Code still stipulates to this day that when the father does not recognize the child until after the child's day of birth, the child's surname can only be changed from the mother's to the father's name, or a combined name, if both parents agree.
https://www.legifrance.gouv.fr/codes/article_lc/LEGIARTI000043895566
Given all that, I wonder how Nicole Coste changed Alexandre's surname to include, or be only (depending on which interview should be believed), Grimaldi, as it was not possible for her to do so without the Prince's consent under Monegasque or French law. Since she and her son have lived for a long time in London (thirteen years, according to the Tatler article), my supposition is that she changed his name under British law, which allows parents to give their child any surname they please. But I wonder if that change is recognized in France or Monaco.
Referring to the earlier discussion about legitimacy under French law: as far as I can tell, the French Civil Code (see the link above) has eliminated all references to legitimacy and illegitimacy, so it is not that Alexandre is legitimate under French law, but that French law does not acknowledge the categories of legitimate and illegitimate any longer.