There’s nothing special about churches. The ACLU has the same tax breaks and associated restrictions.
It’s why the ACLU is actually 2 different organizations. There’s the ACLU and the ACLU Foundation. ACLU is a political lobbying group that is directly involved in political activities and therefore has to pay taxes and donations to it are not tax-deductible. The ACLU Foundation other is a legal services charity that is tax-exempt and donations to it are tax deductible.
Churches simply don’t have the direct political arm. If they do, then they should be taxed. I once saw a church lose it’s tax exemption when the pastor told the congregation they should vote for Obama.
If churches should be taxed then all non-profit organizations should, and you therefore shouldn’t be able to make any kind of tax-deductible donations.