You can have two different accounts open at any time or more on the same PC. I often have one on one screen, one on the other open.
The thing is

*Must have 2+ accounts
*Must click the [play] button before opening a new launcher or it will say client already open
*Datacentre and other stuff does not matter

There were some confusions with early legal terms when you buy PS3/PS4 and it makes you think you can sign in to a different character on same account but this is wrong, what sove92 is probably referring to is that you used to be able to crash your client so that your character 'is still logged into the game' then log into another and freiend yourself so you could send mail between your characters on the same account. You could not physically play both at the same time.