USB-C dock with HDMI graphics on Late 2013 MacBook Pro
USB-C is a connector, not an interface.  So, on your 2017 MBP, the "USB-C" ports are actually Thunderbolt 3, which includes displayport over USB-C.  It is this video signal which is able to drive the monitor connected directly to the dock.

One option would be to get a TB2 dock (e.g. https://www.startech.com/Cards-Adapters/Laptop-docking-stations/thunderbolt-2-laptop-dock~TB2DOCK4K2DP).  They are pretty rare now that TB3 has taken over, and hence expensive.  You could potentially connect that dock either directly to the 2013 MBP, or to the 2017 MBP via a TB3->TB2 adaptor (I've not tested this, and I know the Apple adaptor is pretty fussy about what it will connect to, but it should be fine as the dock is true TB2 and not mini-DP).  The dock has two display outputs - these are DP, but can be converted to HDMI with further adaptors.  It's gonna be pricey, but would allow you to swap the setup between laptops pretty easily.

If it were me though, I would just keep the dock for the 2017.  Your 2013 has an HDMI port built in, and you can readily convert one of the TB2 ports to HDMI with an adaptor.  Your monitors can each handle 2 HDMI cables.  So you could connect both laptops at the same time, and just switch between them using the monitor input settings.  It's not as elegant, but it will be a lot cheaper.