I used 3.5.2 on an iPhone 7 to log in to an NYPL account. I then added another library, Howard County Public Library System. On the account screen the login form for Howard County is locked: the "Barcode", "PIN", and "Log In" controls are greyed out and I can't type in them.

I added a third library (Alpine Public Library) and saw the same results. This is my first time using iOS SimplyE so I thought there was maybe an inconsistency across platforms where on iOS you just can't be logged in to multiple libraries at once.

But then I logged out of my NYPL account (again, this was the first account I added) and it unlocked the two login forms beneath it. I logged in to Howard County; the NYPL and Alpine login forms stayed unlocked. Then I logged in to NYPL again. At this point the third library I'd added (Alpine) had its login form locked. At this point I was logged in to two libraries but not able to log in to a third.

Logging out of NYPL unlocked Alpine's login form.

So it seems like we have a problem where you can have any number of libraries in your account list, BUT, once you log in to the first library in that list, you can't log in to any of the other libraries in the list. This only seems to apply to the first library in the list: if you log in to the second library you can log in to the third and the first. But if you log in to the second library in the list and then the first, you can't log in to the third.