Can customers choose between several of my shops for pickup?

Yes. Every store location with pickup turned on appears in the Choose a store list at checkout.

1. Go to Inventory → Store Locations (or click Manage store locations on Plugins → Click & Collect).
2. Click + Add for each branch, or Edit an existing one.
3. Fill in Store location name, choose a Territory and Sourced from warehouse (add them first under Inventory → Territories and Inventory → Warehouses if the lists are empty), then the address, Pincode, City and so on. Turn on Pickup point and Active.
4. Click Save.

Only locations with both Pickup point and Active on are offered. Without the Multi Warehouse & Store Management plugin you can add one store location; that plugin lets you add more.