There are a couple of problems here.
One is that the 13" M1 MacBook Pro only supports a single external monitor.
MacBook Pro (13-inch, M1, 2020) - Technical Specifications - Apple Support
The other is that even if you had a Mac that supported two or more external monitors, you could not hook up both of them using that dock.
The HP USB-C Dock G5 looks like a plain USB-C dock with three places to plug in displays: two DisplayPort ports and a HDMI 2.0 port. I see no indication that this dock uses DisplayLink or other second-class workarounds – and thus my guess would be that it drives three displays using DisplayPort MST.
Macs don't support MST and only support driving one display from a plain USB-C dock, or two from a Thunderbolt dock. A typical symptom of using a Mac that supports multiple monitors with a dock that uses MST – and plugging two displays into said dock – is that the Mac only sees one display, while both displays seem to always run in "mirror" mode (since the dock is feeding copies of a single video signal to both of them). (In your case, there would only be one video signal even if you were using a Thunderbolt dock.)
I'm not surprised that installing DisplayLink Manager had no effect. That software only works in connection with an external "stunt box" that contains a corresponding "magic decoder ring" chip set. Neither a "stunt box" alone, nor the software alone, will get you any video outputs.
