Hi @iamlost,
This sounds like you are using a Private Address to Connect to your new network. This randomizes your mac address so you cannot be tracked from network to network using your device ID. While this is a great privacy feature for public wifis, this also results in complications when connecting to your regular home network.
On your iPhone's Settings, navigate to the wifi settings and tap on the network that you are connected to. There should be an option for "Private Addressing." Toggle this setting, restart your iPhone and see if the client name is the same in the router settings.