Praise be to Allah.
If a person owns wealth that reaches the nisaab (minimum threshold) and one full year has passed, he must pay zakaah on it, whether this wealth is cash, gold or silver, from the mahr (dowry) or anything else, whether he is saving it for a trip or to build a house or any other purpose.
The nisaab is the equivalent of 85 grams of gold or 595 grams of silver.
Based on that, if your mahr that is kept in the bank is equivalent to the nisaab or more, then you must pay zakaah on it every time one hijri year passes, so you must pay one-quarter of one-tenth of it (2.5%).
We must also point out that it is not permissible to keep money in riba-based banks except in cases of necessity or fear, to protect it, if there are no Islamic banks, so long as there is no interest involved.
And Allaah knows best.