You need to link your accounts. The whole process is very unintuitive, and took me a while to figure out.
First you need an Asmodee account.
Then start Pathfinder on your PC, link it to your Asmodee acount, then click on the link in the options, then click the iOS one.
It will spur out your account idea (careful, no O there, only zeroes) and a code you have to input in your iPhone within 5 minutes.
Go to your iPhone, and this is the tricky part: do NOT login with your Asmodee account there! Just get to the link setup page, input the credential you got. If all went well, your Gamecenter ID will be linked to your Asmodee account, which will be linked to your Steam account, allowing sharing of your files all around. You won't have to login ever again after.
You can only do this once per Gamecenter/SteamID, so be sure to be logged in with the correct one before linking. (I may have linked my wife's here, whoops!)