I installed kdocker, docked application and it worked ok. I added kdocker to system settings autostart. But after reboot I didn't see kdocker icon in system tray but only this kdocker pointer to choose application to dock.

I want to have after each system start kdocker icon visible in system tray so that when I open a program that is assigned to dock I don't have to make docking procedure again, but only click on kdocker icon to minimize application to system tray. What are the proper steps?

Is kdocker still a thing? Last I knew it was a third-party application from like 15 years ago, not made by the KDE Community, so maybe asking the author of it seems a good idea if there is indeed a recent release.

Yeah. But is there any alternative to have this docking effect?

I hoped it might be implemented in kwin, but apparently not:
https://bugs.kde.org/show_bug.cgi?id=421682

Maybe such feature could be implemented in system tray, not kwin?

FWIW: the reasoning in the wish, while a bit sparce, is accurate, as the System Tray is not a docker. Did you check out Latte Dock instead?