I am not an expert but here is the advice I would offer you (as always, housekeeping requires that you investigate how this applies in your case before doing or saying anything that might prejudice your position).
Please correct me if I’m wrong, or merely misunderstanding, but I believe you are speaking of someone, who might be yourself, who has already dual nationality, (perhaps mixed parents, or parents in one country / regime whilst born and raised in another. There are many ways in which this can become effective).
‘At the time of seeking asylum’
Perhaps the person has since determined one or another nationality, definitively?
In France, if you are seeking to become naturalised, a legal French resident or a permanent citizen with the same rights as a French person, you need to decide your status.
Flitting between being a guest ‘arbeiter’ in Germany, for example, and a seasonal worker in France, with tax coding, insurance and temporary papers, is only a protection against being required to leave the territory, while you are actively employed.
It doesn’t confer you the rights of citizenship or social security, but there are protections under the Geneva Convention and European Law, that all EEC members must comply with.
However, each member country can apply its own laws, where these are in conflict with European Law.
For example, equality of work opportunities, in principal, is frequently disregarded when the choice is between a foreign or a French applicant. Thus the work open to immigrants is more arduous, lower paid in spite of their high qualifications or skills, and jobs with higher pay and better conditions are offered to native workers.
If you are discovered to be pursuing two applications for identity papers and status under two nationalities, you could find yourself in the predicament of being sent back to your point of origin on entering the EEC. If this were Greece (Lesbos) Italy or Turkey, it would be catastrophic.
The applicant needs to decide, here in France, upon their definitive national origin or identity, (which one) and under which regime or country they will pursue their claim.
No chance to ‘hedge your bets’, or maximise opportunities for one or other countries to accueillir this or that nationality.
Even for EU citizens, if they claim French nationality, they are expected to renounce their own original nationality.
I don’t know about the other country’s national Social Security rules and regulations, (if you speak the language, you can research online using their government site, or copy paste and use Google translate to discover them).
Under the Social Security system here in France, effectively they (the SS) can clamp down very quickly if they suspect someone of abusing the generosity of the system put in place to support asylum seekers.
These are people in desperate need, fleeing for their lives from political turmoil, war, persecution or natural disaster. They take a dim view of ‘economic migrants’ attempting to drain France’s resources because they’re better than at home.
So, to answer your question, the person in that situation would need to evaluate their position in each country, their place in each society, their links to each of the two nationalities they are claiming, and decide where the advantages outweigh the disadvantages.
Other countries may be more flexible about keeping the dual nationalities whilst allowing a work permit and settled status. But their benefit system may be more rudimentary, there may be a higher level of independence required, ability to turn to different work ethics, resourcefulness, hard work, and yet, comparative freedom, more social diversity, opportunities, living there.
France has a very evolved social system, and it is hard for outsiders to prove they can be part of it. The requirements and rocedures can go on for years. There is excellent healthcare, however, education, social housing (eventually) , food banks and work opportunities.
But status is all, there is very little social climbing in France, particularly by , and once here, integrating into the wider community is sometimes difficult, for the children of not for the parents. There is also the danger of children being taken by the state if they don’t fit in easily.
Discussing these points with family or friends is advised (approach the relevant authority once the decision is reached).
Keeping one’s own counsel when working with a refugee agency is to be advised as once things are said they cannot be unsaid.