Let's assume I write a program and release it under the GNU GPL. Can I call this software "GNU software"? If not, what is necessary to make it a part of GNU?
No, it does not; releasing it under the GNU GPL makes your program free software.
The GNU project has a web page outlining what you need to do to offer your software to be part of the GNU project, which might be summarised as follows:
Making a program GNU software means that its developers and the GNU project agree that “This program is part of the GNU project, released under the aegis of GNU”—and say so in the program.
This means that you normally put the program releases on ftp.gnu.org.
It means that the developers agree to pay attention to making the program work well with the rest of the GNU system—and conversely that the GNU project will encourage other GNU maintainers to pay attention to making their programs fit in well with it.
Another important GNU standard is that GNU programs should come with documentation in Texinfo format. That is the GNU standard documentation format, and it can be converted automatically into various other formats. You can use DocBook or any other suitable format for the documentation sources, as long as converting it automatically into Texinfo gives good results.
If a GNU program wants to be extensible, it should use GUILE as the programming language for extensibility—that is the GNU standard extensibility package. For some programs there's a reason to do things differently, but please use GUILE if that is feasible.
A GNU program should use the latest version of the license that the GNU Project recommends—not just any free software license. For most packages, this means using the GNU GPL.
Note that a copyright assignment is not a requirement:
For a program to be GNU software does not require transferring copyright to the FSF; that is a separate question. If you transfer the copyright to the FSF, the FSF will enforce the GPL for the program if someone violates it; if you keep the copyright, enforcement will be up to you.
The GNU-project is a project to develop a free UNIX-like operating system (read also). They developed the GPL to license the software their operating system consists of. But that doesn't mean every GPL-licensed software is part of the GNU-project (and hence GNU software).
If you want to make your software part of the GNU-project, you should read what they have written about how you can do that.
No. GNU is an organisation, who publish their license the GPL. Just like you wouldn't call your software "Google software", you can't call your software GNU software.
You could call it GNU software if you developed it as part of a project for GNU, for example if GNU employed you to develop a new distribution platform for their licenses.