I don't think the extenders support handoff technologies like 802.11k/r/v fully, so by default they end up creating separate SSIDs on the assumption that most devices won't be moving. If you log into the extender's web interface (you can retrieve the IP address of the extender from your router at http://192.168.1.1/ ), you should be able to rename the SSIDs broadcasted by the extender to match.
Some devices will automatically roam in a best effort manner when two different access points broadcasting the same SSID are detected. Other devices may end up being sticky, and will require turning Wi-Fi off and back on in order to join a closer Wi-FI access point. For stationary devices, the only problem I see happening is that a device may latch onto a further access point and not roam, which if the signal is marginal to that further away point, will result in a poor experience. Either way, worth changing the names to match and see how it works out.
As for the network extender, it's creating a separate "SSID" as it's called, but it's likely not creating a separate network. If your devices can maintain the same IP address whether they are on the router or your extender, that would be the same network. If IPs change (such as 192.168.1.10 to 192.168.2.15) then you have a different network.