List their yearly monetary needs / Present them to management for approval
/ Set their salary based on that?
The practice itself, to have different salaries based on circumstances that have little to do with the company or the productivity is very common. As an easy example, I'm sure costs for rent and food differ in Norway from region to region just as in any other country. In my country, for a good salary in a northern town, you would not find a bridge to sleep under in the most epxensive southern town. A salary in the southern town tends to be at least twice as much as in the north just to keep their employees. But what the employees actually have in their pockets after all neccessary expenses like rent, food and utility is very similar between the regions.
There are also instances of companies paying for public transport tickets for example. Which if you use public transport is a lot of money over the year, if you go by car or even bike, it's worth nothing. Or maybe they offer a company car. That is nice if you need a car. If your family already has two in the garage, it's kinda "meh".
So yes, different payment based on variables that are not productivity of the employee are legal and widespread.
It does become a problem at the "approval" step though. For example which region you live in is not exactly covered under the GDPR and is information that is already available to your employer in way more detail anyway (I assume they have your full address). More personal information is not available to your employer and for good reasons. Even just having such information runs against anti-discrimination laws.
It is also not only legal, but legally required in some EU countries to take those things (age, dependents) into account when laying off people. So there can be legal discrimination to protect those most vulnerable. So maybe it's better to create that sounds like a protection for vulnerable groups than a payment scheme. Something like extra days off for specific needs. Like say you not only allow the legally required maternatiy leave, but give 4 weeks on your own paid leave on top. Or give a day off not only for bereavement leave of first degree family members, but also second degree. Or maybe you grant a paid day off for a persons kids first day in school. Although this does need prove of some kind, I'm pretty sure if properly worded that would not be a legal problem.
It would also been seen as more fair by other employees. Giving out lump sums of money for reasons unrelated to the job seems unfair, while giving out benefits directly related to the specific needs of employees seems very friendly and helpful.
What steps would need to be taken to ensure that this process is compliant with the law?
You need to get a lawyer (and maybe a union representative if your industry is heavily unionized) and go through your plan item by item and check whether your company is allowed to have and process this information. And if your company is allowed to discriminate based on this information, even if they have it legally.
And in the end, when you have your watertight legal plan, you have to ask yourself how you will attract talent, when you pay based on their needs. Why would a young, single, very educated person do overtime for you, when an undereducated, lazy slacker next desk earns double what they do. Not because they do a better job, but because they have a wife and three kids and have a mortgage on their house.
So it's legally a very costly endeavour. And it forms your company not as a company of talented hard workers, but of people who have the most liabilities. Because that is what you would be paying for.