Do other religions have power over demonic influence?They could, if the Lord chooses to grant to them, or to a person from among them, this special grace and power. Odds are He has done this before. After all, "some and even very many of the significant elements and endowments which together go to build up and give life to the Church itself, can exist outside the visible boundaries of the Catholic Church: the written word of God; the life of grace; faith, hope and charity, with the other interior gifts of the Holy Spirit, and visible elements too" (Unitatis Redintegratio, no. 3).
However, He normatively exercises His power over demonic influences through the Catholic Church, so we should expect to find this power in Her, first and foremost. If another religion attempts to exercise this same power, we can only hope that the Lord has granted this to them. Unfortunately, we have no guarantee that He has given them this power, whereas we know with certainty that this power has been granted to the Church.
So, do other religions have power over demonic influences?.....maybe