@csaba.szabadkai interesting case, I would suggest to combine the information of the overlapping lines into one line at these locations which contains all info from the overlapping lines. Is that a good solution for your case? 

In case thats a suitable solution I can provide a workbench

I am not exactly sure what you are looking for, but if I am understanding correctly you have lines which you want to shift their entire geometry parallel in which case you could look into the OffsetCurveGenerator.

To me this solution feels a bit tricky but okay, I am curious how it looks like :)
it also depends on how close your  original lines are located, otherwise it can become a mess

I think we may both be taking this question in two different directions. My understanding was that the current/input data is multiple lines that all have the same exact geometry. They are, in a sense, right on top of each other but have different attributes. I agree with your solution that these multiple lines could be combined into 1 feature with the attributes of all. But the way that I was reading the question was that they may have been wanting the lines separated out apart from each other, giving each line a new geometry that is parallel to its old geometry. This way the multiple features would no longer be visually on top of each other, and instead be spread out. Do either of these sound correct @csaba.szabadkai?
The reason I am thinking this way is because I work with fibercabling myself and from an x,y perspective only (no z values) the cables strung between two poles would all have the same geometry. But for the visualization purposes of printed reports, we have offset cabling a set distance. This way if there are multiple cables running the same line, they are displayed offset to one side or the other from their true location so that each cable can be viewed when printed.