Debian has transitioned to 64-bit time_t in 2024. Packages found incompatible were rebuilt and renamed into new "t64" versions. Unless you are an administrator who upgrades the old system, that should not bother you at all, as everything should work as usual (and the new binaries are literally the same as old ones on x86_64).
If you check the control file (package metadata) for libgtk-3-0t64, you can see that it also “provides” the virtual package libgtk-3-0, replaces and breaks it. This combination tells the package manager that for all intents and purposes libgtk-3-0t64 is a substitute for libgtk-3-0. That's why when you try to install/update libgtk-3-0, it checks that libgtk-3-0t64 is there, and says all is good with it. Declaring a virtual package this way allows software packages from external sources stating old dependencies (like Tixati) to continue to work on newer systems.
https://www.debian.org/doc/debian-policy/ch-relationships.html#virtual-packages-provides
Now why doesn't it work for you? This brings us the question of which desktop environment and file manager you are using, and which commands get executed when you double-click the file (I assume you are doing the simple thing). Not all of the tools that can get called to handle the solitary .deb file (dpkg? gdebi?) work with dependencies in the same way apt does. This can bee seen as the last line of defence against people downloading random files from internet and never reading the manual.
The answer to that question is irrelevant in our case, though. You already know that the dependencies are installed — most people running graphical desktop and any GTK-based applications have them, and you already ran the previous version of Tixati. So you can simply install that single package with dpkg by telling it to ignore the dependency checks. The only files it contains are program binary, desktop file, and an icon, so it simply overwrites the old version with a new one, and that's it. Alternatively, you can try something like
sudo apt install ./tixati-xyz.deb
to let the main complex package manager tell you that everything is all right with dependencies. The result will be the same.
