They typically specialised in the metals being cast, but not usually in the types of items being made. Different metals and alloys will need different temperatures for casting, and can need different furnaces to achieve those. There are examples of foundries that used two different bronze alloys (one being a high quality bell alloy) being made with two separate workshops, one for each alloy.
For casting different objects, all that is needed are different moulds. Given a market for cauldrons and skillets, it would make little sense to only make cauldrons (cooking vessels were a major product of bronze foundries, and are prominent in foundry archaeology since their moulds are relatively large). Other items could easily be made too - some (post-Medieval) foundries that mostly made cooking vessels (because there was a large market for them) also made bracelets for export to Africa.
One exception is cannon foundries, which could be established by the state. Their job being to make cannons, they made cannons. Cannons could also be made by private foundries, who didn't specialise, and would also make other products (e.g., bells).
For some details of archaeology of late Medieval and early post-Medieval foundries, see