Hello again and thank you once again for your answers.
I will give you more details about my booking and make things simplier for you.
I made the reservation for two adults. They automatically sent me one confirmation code that refers both persons. About 3 days later they sent me two e-tickets (one for each person) with different e-ticket number each one. Each e-ticket refers all of the four flights (Ath-Ams, Ams-Edi, Edi-Ams, Ams-Ath). When I go to "manage my booking" it showes both names of the persons travelling and all of the 4 flights but it shows only my e-ticket number and the ticket number of the second person is blank.
There is another detail. In the field "Free baggage allowance" it writes "Baggage details could not be retrieved." but I don't think that's a problem. I know that each passenger can carry one baggage with 23kg maximum weight and one handbag in the plane. Am I right?

Now for the ID card. I wanted to know if I can travel without a passport using only my ID card which has both greek and english characters. So I called the british embassy in Athens to ask. They couldn't answer me for sure. Then I called the greek embassy in UK. They told me that my ID card is accepted in UK but I have to call KLM to ask them if they accept to travel only with my ID card and without a passport. I tried to call KLM but I didn't manage to talk with someone because they didn't pick up the line and I am waiting for a very very long time.

P.S. My ID card has no expire date written on it.