That is happening to a LOT of people because of the insanely high numbers of people trying to log in. The Launcher is slow to give people the play button also. Then if there are a lot of people trying to get to the 'LOBBY' from the start screen, you might get the 2002 error. If you get lucky and make it to the character screen, you'll end up in a login queue. And sadly, can even get an error and get disconnected from that, which means needing to load the launcher again. I got the lobby error a couple of times once I finally got the play button. Finally got in to the character screen, had a queue of 2835. Got down to 1783 and got disconnected. Luckily, I got back in and hadn't lost any ground in the queue and just made it in after 50 minutes total waiting, so have hope!

This is all not unexpected, Yoshi said it could happen in his post on Lodestone on 11/29 concerning congestion for Endwalker's Launch/Early access.