Is there a STM-core where hardware-serial enables to assign IO-pins to serial1, serial2, serial3 and works immidiately?

First, you have to be clear about which core you are talking about. There is the official STmicroElectronics Arduino core (recommended) and other maybe abandoned cores such as "Roger Clark (dan.drown)" core.

Second, if it is the Stm32F103xxxx (bluepill) then it appears to be true that the mapping between the UARTs and physical pins is fixed.
Also, I can't see a way of remapping these in the STM32cube IDE

image

Third, there is a dedicated web site https://www.stm32duino.com/ for Arduino STM32 stuff.

I use ST-Link hardware (clone $3 on Ebay) plus STM32Cube programmer software to integrate these with Arduino. There are numerous other possibilities.

This is the tutorial I recommend: Program "blue pill" with STM32 Cores in Arduino IDE · One Transistor