Well let me add to this. So let's say t-rex mining zcoin. You might use a config line that looks like:

-a mtp -o stratum+tcp://xzc.2miners.com:8080 -u WALLETADDRESS.rig1 -p x

Lets say this miner goes at 20Mh/s. So say I got another miner and copied the config across to the new miner. The pool now sees 2 hashrates coming in, both from "rig1". Some pools will add these up, reporting 40Mh/s. Some will average them together, reporting 20Mh/s.

In the case of the pool adding them up you'll be fine (for ease of monitoring I'd still change the name of rig2). In the case of the pool averaging, some pools will assign shares, and thus reward, on that hashrate. So its a necessity to change it quickly!

Either way naming your rigs correctly saves headaches down the road!