I have finished LWS2 with my warrior and tried to replay it with both my necro and mesmer in order to get some achievements done but... no achievements icons or even if I do what is needed for the achievement I don't complete it. An example, I did 5 times the 'Cornered' episode for getting the achievement 'Explosive Avoidance'. Then I hopped into my warrior who also had active the story and through the replay feature I did it again and voila there was the active achievement icon which got in the end just fine.
I thought maybe it had to do that the story was also active on the warrior but no. I quit the story on warrior and again with my mesmer proceeded into the next chapter 'Entanglement' and more specifically the episode 'Discovering Scarlet's Breakthrough' attempting the 'No Tricks' achievement. Again no active achievements icon on the boons bar and even though I didn't use any of the available story weapons (as the achievement requires) I didn't get the achievement in the end.
So it seems like achievements are locked (?) to my warrior (the first character completing LWS2). Is it like this or am I missing something?