Firstly this question is very general there are many electoral systems and many consideration that might have to be taken into account for this to be done in practice e.g. religious observances in some countries.
But in general
First what can demography encompass
Demographic analysis can cover whole societies or groups defined by criteria such as education, nationality, religion, and ethnicity.
Demographics can encompass a lot of things and requires careful analysis As you can see from this quote from Wikipedia (emphasis mine)
so some potential issues
- Quotas could only work in a system where you elect a party not an individual each party would then be obliged to provide enough candidates of each kind so that if they won they could fill the quota, this would be very difficult for small parties any split between parties would raise the questing which party has the fill what quotas or perhaps all the parties do
- Any system that requires this would have to determine which groups require representation and be able to determine what proportion of parliament would need to be in that group. This could lead to 'god' organisation that determines in effect the politics of the next parliament i.e. if we recognize demographic X then they're likely to tend towards Y. Groups that are not given a quota might feel they are not part of the democratic process. Recognition of a group could be seen as giving legitimacy to views held by that demographic
- With or without intra-group diversity requirements individual group discrimination rules (e.g. rules about what men/women can or can't do) could prevent some other quotas being reached
- At a certain point you could end up with a post that very few people could or would be willing to fulfill and therefore a specific person is the only appropriate choice and can not be removed
- if a party stands against a particular demographic expecting them to fulfill a quota could be problematic for them
- Recruitment, some political parties already attempt to hire according to some quotes however a problem faced by them is lack of quality candidates suited to fulfilling these quotas
- There are many groups that require representation but might not be suitable for running the country. Would people be comfortable with for example handing 10-20% of parliamentary seats to those would can't read or do basic math.
- Discrimination rules/laws might prevent the recruitment process from selectively recruiting to fulfill quotas. Parties that have to fulfill quotas bay be unstable due to having to many divergent views
- Determining who is in what demographic group this would be important so quotas could be set and for determining what demographics an individual MP represents
- Over or under representation from a fairness perspective a person who identifies with more demographics will end up being more represented and those who don't identify with many would be under represented
- Visible identification - all demographic groups that you want to have represented you will have to join this would be awkward for some people who may not wish to do so to avoid discrimination
As an additional consideration consider who or what would in effect ends up in power if a demographic has it'a own institutions that hold power the country could in effect be giving power to that organisation. Example, if you recognize Roman Catholics as 60% of the population then Roman Catholics get 60% of the seats and to recognize who is a roman catholic and who is not they are required to be a member of the roman catholic Church this could give the church a certain amount of power of those representatives enough perhaps the the church would have effective power.
TLDR; it's an interesting idea but would be very difficult to implement and could suffer from a number administrative difficulties