Yes. Birgitta Jónsdóttir is properly referred to as Birgitta Jónsdóttir or just Birgitta, or by her first and middle name if disambiguation is required. Jónsdóttir alone is meaningless as far as name goes. Icelandic phone books are ordered by first name.

edit: although to be fair, in the context of an article written in English aimed at a general audience this is not an easy situation. Properly using just the first name might throw off people who don't know about the Icelandic conventions (that would be most of them) who might then consider this use as strangely personal. On the other hand, repeatedly using the full name might look weird too. Personally I'm not convinced these are worth the trade-off of being culturally incorrect, but I'm not the one making the calls.

This appears to have been fixed.

Whatever. Good writers write for their audience. This article was written for an audience residing outside of Iceland.

I read a ton of articles that routinely use a persons first name, why suddenly should proper formality be artificially upheld in context of a culture that uses patronyms? They use first names as names, and in the context of an article they can use a first name without it getting confusing.

But in certain areas and with certain publications this clashes with house style. For example the tabloids/red-tops are happy to refer to Boris Johnson (London Mayor) as BoJo or simply Boris, the broadsheets would say Boris Johnson first and then use "Johnson" to refer to the person as using a first name is considered impolite and/or disrespectful in formal writing (which the broadsheets tend towards at least in non-editorial pieces).

More in depth or higher class pieces that are not expected to be widely consumed may well then go a step further and assume the reader has detailed knowledge of name systems (for example Chinese "Family Name, Given Name" ordering).

So to reiterate the expression of one of the parent posts they appear to be writing to their audience. Be offended or confused if you like.

The reason you'd call someone Mr. Smith in the English speaking world is because surnames are culturally significant. Because they group entire male lineages together.

That's not the case with patronyms, so this usage has never developed. In Iceland you'd never use just the patronym. It's not a proper name, just an indication of parental lineage.