So I have a question about the robot mods. When I store my mods in the workbench, does that carry over into other settlements? or only the settlement I put them in?

Only the settlement you're in until you set up a caravan between settlements. Then, those connected workshops share inventory inside the workbench.

Well I know that it didn't share weapons, and aid items, I was just wanting to make sure it worked like if I have water in x settlement even though its not in the workshop i still can craft at that settlement. So mods are the same?

The way caravans work is they let you craft using materials in connected settlements workbenches, but you can't straight up pull items from those workbenches. The idea is that you conveniently ordered and had delivered all the parts you needed at the moment you crafted an item.

So it's ideal to set up a central equipment cache that you plan to frequent (if you're of the "no fast travel" variety, I recommend Hangman's Alley or Home Plate,) but crafting materials can go into any other settlement as long as you have a supply line to those settlements.

You can have a supplyline go to home plate?