Trust issues with my PhD advisor, is it worth fixing or can I get out? I am a math PhD student considering changing universities.
I have suspicions that my supervisor has not been very honest with me
since I told him that I considered changing universities (and therefore
supervisors) because of shift in interests. He got angry with me, told me that it would bring
bad reputation to the research group and that I can't just leave after
1 year.

In a conversation I mentioned a university, in which there is a professor,
call him professor M, whose work interests me.
My supervisor said that professor M is the only
one in his field in the university (apparently
with the intent of discouraging me from
trying to apply to that university).
I looked up the website of professor M's research
group, which seems to tell a different story
(plenty of PhD students, at least one post doc).
I found this a bit suspicious.
Moreover I recently realized that my supervisor
has coathored an article with an emeritus professor, who
is in the university, who has worked on the same field, although
not part of the research group of M. The "research group" in my
university is not that active itself.

When I told him that I want to consider alternatives
he told me that the doctoral programme is collecting
information about PhD students (which was true), and
that he needed to know whether I was part of the group or
not. He gave me 3 days to decide whether to stay or leave the
doctoral program. I asked the coordinator of the program about this,
and he told me that the list in question changes all the time and
that it is not nearly as serious matter as my supervisor claimed.
It should be noted that my supervisor is notoriously bad at 
bureucratic/adminstrative university matters.

I suppose in all the above cases it is possible that
the explanation is something less malicious, such
as ignorance or misunderstanding.
In any case I am not in good terms with my supervisor
and I don't think we trust each other very much.

The dilemma is that the funding is good and secured, and that
the alternatives are abroad (and uncertain).

Could the situation be worth fixing?
Should I run for my life?
Is there a risk that if I try to change universities
he will refuse to write a recommendation letter or
write a mild one (he supervised my Master's thesis)?
Faculty are human beings. We might be given only one slot a year for a doctoral student to work with us and we thus invest (or we feel we invest) an incredible amount of time into the students in our lab. We aren't paid extra for this, which perhaps exacerbates the feeling of ownership. 

While I don't know you or your advisor, I would find it easy to believe that his feelings were hurt by your desire to shift PhD programs. You are basically taking his investment in you (which you may feel is minimal, but he may feel is considerable) and throwing it in the trash. 

Managing people's feelings are part of being a professional academic. You might as well practice being good at it.  Some day you'll be on the other end of the equation -- being told by a student that you had high hopes for that they no longer value working with you and that they want to move to a different program altogether. It's all part of the karmic cycle of being faculty. 

Now, with that preamble out of the way. You need to let your advisor know that your shifting programs isn't seen as any rejection at all of his lab or his style of mentoring you, etc. etc. (even if part of this might be true); but rather that you are seeking something different at the other program. It's the stereotypical "It's not you, it's me" form of a break up.