When running several
bitcoind nodes on different machines, can I simply keep the same
bitcoin.conf file everywhere, or do I need to ensure that a different
rpcuser is set up for each?
The rpc user/pass is only needed if you're planning to access the node via RPC. If you're running Armory on top of it, for example, that requires RPC and hence you would need it (Armory auto-creates the rpcuser/password for you, though).
If you're running a node and don't need the RPC functionality, no, you don't need those parameters set.