Aid items are shared. I can personally verify this. The issue is that you interpret "shared" to mean "dumped into a container that can be accessed from any workbench", which is not what happens. Items can only physically be pulled from the Workbench at the settlement where they're generated. This applies to all shared items, including Junk. If Settlement A generates a telephone and Settlement B doesn't, you can't pull the telephone from the Workbench at Settlement B.
What the Supply Lines do is cause crafting stations to check the inventory of all connected settlements when deciding whether or not you can craft an item. If Workbench A is missing 2 items, but Workbench B has the missing items, then the crafting station will let you create the item.
Think of it this way: you're trying to make something, but not all the materials are there. However, you know that the missing materials are at another settlement. You turn to your Provisioner and ask him to pick up the missing items the next time he's there, and he agrees. A few days later, he shows up and hands you the stuff you asked for, which you immediately use to make the item you were originally trying to make. The game just cuts out the inconvenience of having to talk to your Provisioner and wait several days, and jumps directly to the part where you can craft the items.