Yes, some people are known to have allergic reactions to silicone. You may well be one such person. It would be advisable to see a doctor to find out.
As to replacing the band, it depends on when you bought the watch and band combination. If within 14 days of purchase you can return the watch and band to Apple and choose a different combination. If you are past the 14 day return window, just buy yourself a different type of band. You don't have to buy a band from Apple. You have loads of options (thousands, actually) from sellers like Amazon.
If you are one of the rare people who are allergic to silicone, don't buy any band with silicone in it. You could try leather or metal or canvas or nylon as alternative materials.