Basenji, looks to me like the Unbuntu repo is the problem. Ubuntu, when they make a distro, just take most things from a snapshot of Debian Testing. And Testing being what it is, sometimes it has problems. Ubuntu certainly test what they consider important features and make some fixes if necessary but I suspect that Lazarus is not on their 'important' list.
In this case, you need to install libgtk2.0-dev and its not obvious to me why its not possible, I have a 22.04 VM here with it installed (easily). And you need install libpango1.0-dev and it needs (eg) gir1.2-pango-1.0 but seems the official Ubuntu package is 1.50.6+ds-2ubuntu1 - that would indicate Ubuntu made a change, renamed the package but did not think to patch gir1.2-pango-1.0 in the same way. In most cases, you can solve such problems by manually installing (eg) 1.50.6+ds-2ubuntu1 - it will then, in deb speak, 'provide' gir1.2-pango and libpango1.0-dev will be happy.
So, in both cases, drill down, manually install what is available from the lower levels up and that usually fixes everything. Remember the package name and the things the package provides can and usually do have different names. Every time the package is patched or even re-packaged, the package name changes.
Underlying problem here is GTK2 is becoming less and less important to the distros. You could, if necessary, install fpc and fpc-src (getting an up to date compiler, 3.2.2) and then build a current Lazarus Qt5 version from source. Should not be necessary but sure would work. Note that building a GTK2 Lazarus from source will not help you if you have not yet solved the libgtk2.0-dev issue. (Putting that aside, building Lazarus from source is always a good idea.)
Don't give up, try those bottom up manual installs and tell us how it goes.
Davo