Reading the label or instructions could shed light. I'd say more info is needed regarding the model size and rating of the switch. Larger switches would likely be expected to have multi-port lugs for parallel conductors, smaller ones not so much.
It sounds like you are not paralleling conductors but are branching to two different circuits. Why? I've seen this done with taps to multiple loads, which is okay, but pretty old-school. Why not use a panelboard?
Whether you need fuses probably depends on whether you already have overcurrent protection upstream.