For those of us who have lived here for a long time, we have had all the rights of any other EU national. A freedom to live as we choose within the law, the right to travel around 28 countries, the right to work here, the recognition of our qualifications, cooperation to ensure we can all receive medical care, the security of shared information on criminals, etc. the list goes on. Now due to an ill informed referendum when many were told that they could have everything they desired at no cost, many of these rights are to be withdrawn.
It was never necessary to have a plastic card or piece of paper to say that you legally resided here. When I arrived in 2007 I inquired should I get any proof that I was legally here. It wasn't required, a UK passport was complete proof of your rights. Join up with the medical service, let the town hall know that you were here, and receive your council tax bill.
Due to the result of the referendum the situation has changed. If you wish to leave the EU for travel, and go to the UK or Botswana, then you will have to obtain proof of residence when you wish to come back to your own home. As there was no uniform system to deal with this it has created a huge problem for the authorities. However they have produced a new system, with websites set up, details of what you need to prove that you do live here, and meetings set up all around the country with English speakers to help make the process as simple as possible.
Compare this with what is happening in the UK. A system for registering that you are an EU citizen is still being developed. Already that has run into trouble by not working if you have the wrong smartphone. A cost for a family that is expensive, and is just another money making exercise. At least the French have a different view. Why should people have to pay to have their rights removed, That is why to obtain your proof of residence here is free.