Those who want to follow a new chain will be able to do so, but if they try to send alternative bitcoins to the main chain these would not be recognized as valid.
In fact, this is a way to distinguish between "real" and "fake" bitcoins and is similar to what happens in centralized payment systems.
The USD chain can be split if a state adopts the dollar and then creates its own local currency.
The sharing of the name, even if the same "chain" is not initially shared, often happens in the monetary sphere: think of the Zimbabwe Dollar whose value is not even remotely comparable to the US Dollar, or, without going to the other side of the world, to the Canadian Dollar, which is worth about 0.76 USD.
Therefore it is easy to recognize different dollars, but it can be complicated to distinguish fake dollars even if they always seem USD: with Bitcoin this cannot happen because any false bitcoins are simply not accepted by the network.