Is this essentially the so-called Repeater Bridge mode?
Sort of. You are right that a Repeater Bridge mode allows both wired and wireless clients to join a router which is linked via wifi to the primary router. The difference here is that the Repeater Bridge mode uses the same interface (or NIC) to connect to downstream clients and to the upstream router, so that it is receiving for half of the time, and transmitting for another half of the time. This can be achieved also with a normal pc, but most routers cannot do that.
Also, it is not clear from what you say whether clients of the 5GHz wireless are in the same subnet as wired clients: if they are, then you have some kind of bridge between LAN-side ethernet and wifi, otherwise you surely do not. So, once again, you may have a repeater bridge router, but then again you may not.
SSID
I will side with Temujin here: giving the same name is probably inconvenient. But I will also add that it might be misleading, if you do not have a bridge between LAN-side ethernet and 5GHZ wifi, because in this case clients of the main router's wifi, and of the secondary router's 5GHz wifi will be in different subnets. You would then have an impossible situation with static IPs, and possibly some difficulties even with DHCP clients.
And if it could work, how do I know which router my device is connected to? Do they automatically switch to the stronger signal?
No, wifi clients do not automatically shift to the stronger signal. And, if you do have a bridged repeater, it would be more complex to find out wich router you are connected to. In this case, you would have to check the MAC address of the AP, with a command like (in Linux):
sudo iw dev wlan0 scan which gives you all relevant details.