Is it possible to continue a PhD program in a different university (PhD transfer)

In the USA it is a fairly common practice though normally a student follows his adviser to another university rather than runs away from him. I'm not so sure about Europe (the rules of the game are somewhat different there) but in any case you should make a few inquiries: first, you should talk to the graduate program director (or someone equivalent) at the place you intend to transfer to about whether they currently accept transfer (or any) graduate students and on what terms (the answers may vary dramatically from one university to another) and second you should contact a potential adviser to make sure that it is fine with him/her (I guess you do not want to start it all over again because otherwise you may just as well wait 3 years where you are, and not everybody would be interested in reading and correcting a thesis on the topic given and supervised by somebody else). Normally, if these two key people give you the green light, the rest is fairly routine (again, in the States).

From what I have seen in Europe, and in Germany in particular, PhDs are treated like jobs. So normally, to do a PhD in Germany, you'd have to find an open position and apply to it. If the research description is close enough to what you've already been working on, it may be possible to transfer progress from your current project. But then you'd be at the whims of a new PI.

Germany universities do technically allow people who are not currently employed in a PhD position to complete their doctorates thorough the university; but there are usually special circumstances underlying these situations. For example, I have a dual-degree, awarded jointly by a German and an American university. Although I had been employed as a researcher at the German university, I never held a PhD position there, and I was only enrolled as a student on paper when my degree was awarded. This whole thing was only possible because my American and German advisors knew each other well and jumped through lots of paperwork hoops for me. I also know of a guy at a German university who had started off in a PhD role, transitioned into a sort of limbo-researcher role, and then a long time later (10 years maybe? whenever funding ran out) actually completed his degree and moved into industry.

Moral of the story: while want you want is technically possible, I would say it's highly unlikely unless you already have connections or a perfect open position in mind.