I said all of that to ask this question....does one have to compile the Lazarus project on EACH Linux distro in order to get it to work?
No but with some caveats (which probably makes the no answer a theoretical one unless knowing what you're doing).
The host platform must provide the dependencies that your application depends on. E.g. if you compiled your program for QT5 widgetset then it will ofc not run on a host that does not cater for that dependency.
Furthermore, wayland vs X11 can be an issue and in a similar fashion the version of libc which was build against.
The versions of the libraries that you build against are important (the host must be able to handle them).
Other than that, in theory, you should be good to go.
A GUI application can be invoked from a terminal and usually provide an error on what is missing. Detailed information can be obtained with strace.