iOS VPN On Demand with multiple VPNs
No, as of iOS 10.3, the VPN won't connect on demand, if it isn't selected in the VPN settings. Obviously, a non-selected VPN setting is meant to be completely inactive, including its on demand configuration. 

I experience the same problem, and am hoping for this feature to be in a coming iOS version, but I am not too positive about it.

However, this answer applies to multiple VPNs of the same type. It seems that if they are of different types, multiple VPNs can be active and even connected at the same time, as described in this question.