It is not only Chasidic. It was a long standing German custom until relatively recently that the chassan and kallah should be seen publicly holding hands after the chuppah. This is to clearly illustrate that they are now married. Although it was not universal in some German communities the chassan and kallah danced together for the last dance of the evening.
As far as tznius it was felt that this is appropriate for a wedding but not for elsewhere. The chassan and kallah never hold hands publicly again.
It is similar to the idea that one should compliment the kallah on her beauty at the wedding, but it would be considered not tnzius to do so at some other time.