As I understand it, the distinction between man_made=bridge and a highway with a bridge=yes, is that man_made=bridge represents the outline of the whole bridge structure, which hosts the highway but possibly other stuff (like a cycleway, pedestrian sidewalks, etc).

So, if this solution is used simply to make the bridge have two names and the outline is arbitrary, then it violates the "one feature, one element" principle and alt_name is a better idea.

But it is a good solution if the man_made bridge represents the actual outline of the bridge structure.

Here is a reference example:

https://www.openstreetmap.org/way/682150936

