Hi ferencfromleanyer,
Thanks for coming to Apple Support Communities for help today with Photos. It sounds like you want to open a previous library on your new Mac, but receive a message that Photos needs to be upgraded to open it.
It sounds like both machines have macOS Mojave installed. Have you verified they are both 10.14.6?
If you open Photos and use the drop down menu and select About Photos, what version is showing on both computers? They should both be 4.0.
If you open your Pictures folder in Finder, do you have more than one Photos library? If you do, you can hold down the Option key while launching Photos to select another library.
If everything above checks out, you can also try repairing the library: Repair your library in Photos on Mac
Let me know further details.
Have a fantastic weekend!