I find the Grauniad article linked by Carol slightly misleading. As always the media attempts to distil a lot of quite complicated information into a simple narrative. In doing so the condensate becomes so dense that it largely obscures the information behind it.
Up to the date that the UK leaves the EU, non-UK EU citizens live in the UK and UK citizens live elsewhere in the EU on the basis of Directive 2004/38/EC. The Agreement therefore has to use that as its starting point. The date when conditions change from the Directive to the Agreement is the date that the UK leaves the EU, currently 29 March 2019. Under the Directive there are four broad categories of people:
1 Those who stay in the UK for less than 3 months under Article 6
2 Those who work in the UK under Article 7, Paragraph 1 (a)
3 Those who are self supporting under Article 7, Paragraph 1 (b)
4 Those who study in the UK under Article 7, Paragraph 1 (c)
Categories 2 to 4 can be divided into two main groups:
A Those who have been in the UK for less than 5 years. These have right of residence as long as they satisfy the conditions in Article 7
B Those who have been in the UK for 5 years or more. These have permanent right of residence under Article 16
All these categories and both groups are dealt with in the Agreement. The Agreement basically works on the principle of maintaining the rights citizens have on the day of Brexit but that might be achieved in different ways for different categories. The agreement also allows citizens to change status.
The article states in the first line: "The registration of 3 million EU nationals for “settled status” in Britain...". The figure 3 million implies ALL EU nationals in the UK will register for "settled status". It goes on to say: "Those who qualify for settled status, giving them the indefinite right to live in Britain, will have to demonstrate five years’ continuous residence and pass a criminal record test". This implies only those in Group B will apply for settled status. Which is it? Really the UK needs two terms, one to describe those in Group A and those in Group B. Also the Agreement does not mention a specific fee for applying but simply states that it should be no more than the fee for issuing similar documents. Whether a passport is a comparable document is open to debate.
The article mentions nothing about changes of status. For example someone might arrive in the UK on the day before Brexit intending to stay for 3 months seeking work. Under the Agreement they would be allowed to continue seeking work and, if successful, would then be allowed residency beyond 3 months under category 2 above. Similarly on the date of Brexit someone might be in Group A and be allowed to continue residency as long as they satisfy the conditions. At some point they could achieve the status of Group B and be allowed permanent residency.
Obviously the converse is also true for UK citizens living elsewhere in the EU, e.g. Greece. As far as I can see from reading the Agreement if someone is in Greece on the day of Brexit as a visitor in category 1 they can continue to stay. Once they reach 3 months and wish to continue to stay they should still be able to apply for a Residence Certificate provided they satisfy one of categories 2 to 4. If they then reach 5 years residency they should still be able to apply for permanent residency. I cannot guarantee this interpretation is correct. Read the Agreement and make your own mind up.
Those of us with a permanent Residency Certificate should be able to change it to whatever new document applies in Greece free of charge with only a criminal record check. If anyone qualifies for a permanent Residency Certificate but doesn't have one, now might be the time to get one? It is also worth noting that under the Agreement a Residence Certificate isn't the only proof of residency and other documentation might be used. My experience of the Greek process is that they are only interested in a Residence Certificate. I had trouble getting a permanent Residence Certificate because I had let my 5 year Residence Permit expire. I could prove continuous employment in Greece through tax records, insurance records and ΟΑΕΔ registration for well over 5 years but they weren't interested. Eventually the local sergeant relented and issued one but I think that was personal discretion and couldn't be relied on in another situation.