A first-pass interpretation of the question might be that the professor is either (1) trying to get credit for a collaboration for which he made introductions but to which he did not contribute, or (2) is disappointed that a student at his university would write a paper with his collaborators without consulting him on the topic. I gather from the question/comments that the O.P. seems to believe that situation 1 is the correct interpretation. If that's the case, then there is probably no need to include the professor on the paper, but the matter is probably best referred to your academic advisor or committee chair. As a grad student, it is usually a bad idea to have these kinds of arguments with other professors without your advisor's guidance---partly because keeping your advisor in the loop protects you and gives you ammo, but also because you don't want to risk burning bridges that may affect your advisor as well.
That said, I find scenario (2) plausible---that the professor is upset not about being excluded from this paper specifically, but is more generally disappointed that a student in his own department would do a project on [x] without consulting the department's own resident expert (him) on [x]. The O.P. claims that it is the collaborators who are the experts on the software in question, but it seems fairly unlikely that the professor is not a researcher in the general topic area related to the software considering that he started the collaboration. The professor may not be as much of an expert as the other authors, but it is still unusual and rather cut-throat to publish a paper on a topic that is central to the research agenda of another professor in your department using software that he had some hand in producing and to not even consult that professor about it. Were a student to do this to me using software I had helped write/inspire, I would be rather hurt by the exclusion for personal reasons and might ask to have a conversation with them about that.
Finally, I have to add that some of the wording in the original post makes me uncomfortable. The OP states that the professor "established a collaboration with another university" and "does not own the product, but is one of the collaborators with the university." This seems to deliberately suggest that the professor is broadly connected to the university while avoiding the question of whether the professor is not involved in the project whatsoever. Yet professors do not, in my experience, have collaborations with universities but with labs or researchers. I feel that the question obfuscates the professor's actual role in the software (also note that the OP has not answered a comment asking to clarify it as of weeks before this answer was written). Obviously this all depends on facts I do not currently have, but I find it ethically hazardous (if not outright immoral) to collaborate on a paper with some of the authors of a piece of software while excluding one author who is also in your own department/school. If this is the scenario, then I would suggest that the OP apologize and strongly consider whether the software used was important enough to the project and conclusions to merit a position on the paper's author list.
Consider: academic software is often very difficult to produce, yet simultaneously the rewards of such software often go to those who use the software and not those who write/produce it. Academic journals, for various reasons, don't tend to want to publish software-related articles, even when the software is useful or even critical to academics. Those journals that do publish software articles are rarely considered top-tier. Scientists will often say, "Fred made some really amazing contributions to science (using Wanda's software)!" but rarely "Wanda's software is a really amazing contribution to science!" I have been on both sides of this in my career and can easily name a dozen pieces of software in my field that are absolutely critical for standard operations and experiments but that rarely get cited. I've watched many professors and graduate students leave academia for industry data-science positions after years slaving over critical software only to be forgotten on a colleague's paper that could not have been written without their contribution.
In short: I can't, without more information, say what the ethical path forward is, but I can strongly advise that the OP and anyone who uses academic software carefully consider whether they are being unfairly biased against recognizing those scientists who write the software on which they rely. In my experience, spreading credit around generously rarely has any significant cost in science, and hoarding credit on the grounds that someone was only involved in writing the software you used bears a high moral hazard.