It is illegal throughout the west, often monitored and harshly punished. As has been demonstrated time and again for decades, including by recent nobel laureate Claudia Goldin, the wage gap is due to women's choice, most notably motherhood. That women are paid less for the same work is false, a myth.
The average wage of women starts falling behind that of men at around the age of 30-something, which does match the point when people have kids nowadays, especially women with higher education. Funnilly enough, in average women earn more than men up to that point because more women get degrees than men.
So yeah, it's related to childbearing.
However...
It apparently depends on how long a woman stays away from work during the parental leave period: the longer that happens the bigger the negative impact on a woman's career and hence her on lifetime earnings.
Part of the reason why mothers stay much longer at home on parental leave than fathers are indeed personal choice, but part is cultural (i.e. societal expectation that the mother takes care of the children), part is legal (different legal lengths of parental leave for men and women) and part economic (insufficient provision of affordable kindergarten places, at times making returning to work more costly than staying at home, which associated with the other parts means women are the ones that more often end up staying at home for years after childbirth due to this).
Whilst the first 2 aren't really something governments can do much about, the other 2 are.
A handful of countries have gone ahead and done things to change this, for example with free kindergarten places guaranteed for every child and parental leave which is just one big gender-independent block of months that can be divided between parents in any way they see fit.
So are these female politicians in Iceland pushing for the real world solutions that can start fixing the roots of the problem or are they just endlessly posturing about the symptoms for image management and political gain amongst the majority gender?!
Women are paid less for the same work. The gap is around 10%. For all men and women in full-time employment not in the same job it's 20%, while the average total difference is 40%. Also you just linked to an article about a person recieving the Nobel prize instead of anything actually supporting your claim.
I guess I should have looked for links a bit more, that nobel has talked at length about it in many interviews. Anyway this myth has been debunked many times and there are plenty of sources even if you don't make an effort, here is another one.
Even if they limit the statistics to the same sector and full-time jobs, there are still many variables, things like men tend to work more overtime and worse hours. If anything, those who defend this myth should provide evidence that this happens, it makes no sense to defend that it is systemic but only be able to back it up with vague statistics from which they draw whatever conclusions they want.
To give an example of facts in reality, in spain where controls are very hard, there are about 20 convictions a year for wage discrimination, and it is not even only towards women. Purely anecdotal.