Hi Jakub,
I also encounter problems with my MacBook Pro. I managed to install and make DisplayLink work but I am not happy with the outcome.
macOS Catalina 10.15.4
MacBook Pro (16-inch, 2019)
2.6 GHz 6-Core Intel Core i7
16 GB 2667 MHz DDR4
Intel UHD Graphics 630 1536 MB
with US Keyboard
I tried the latest and previous official version of app from the page version 5.2.4 and 5.2.1 respectively. After I enabled permissions in Security and connecting display cable to external monitors nothing happened. I tried it on multiple docking stations. My port are also working fine.
After that I tried various BETA versions. I used one which you pointed in this thread, version 1552. After installation and enabling I managed to get it work. However it's possible to use this functionality if I explicitly activate DisplayLink Manager. After opening app I can see a shortcut in Menu Bar and external displays starts working. Keep in mind I am not able to do that with official release. However I can see that some of my colleagues managed to have official DisplayLink working with their devices. In their case they also don't see an icon but everything works fine.
So my problem is almost solved with one exception. When I close a lid 3rd, built-in display is also available and I have to rearrange the order and set the primary screen. What's worse, when I am in this state after around a minute all screens go black and they turn off. Even when I am moving a mouse pointer or type on a keyboard. My energy saving parameters are set high enough.
I tried also an Amphetamine but that didn't help as well. Every time a screen goes off.
I can even live with open lid as I use fingerprint sensor. But there is no point in having built-in screen turned on all the time
