I have lots of repeaters in my home (actually known as range extenders) and ALL my devices can connect to them, because they are all connected to the same network! Meaning, the range extenders must be connected to the main router's same SSID. If my router signal's name is Mike's Router, the extenders must also have that same name. Otherwise, one does not have an extender/repeater, they have a separate AccessPoint (AP) that is connected to the other router, but is not the same wi-fi network, it is on it's own network with a different name.
Yes, a second router can be a repeater, but it HAS to have same SSID name as the main router. So, dig into your second router manual and figure out how to setup it as a repeater, not an AP. Google is your friend here. Btw, my range extenders can be set up in extender mode OR in AP mode, and I have some with both modes going. Routers can do the same.