There are several other tax-exemption provisions which do apply to political organizations, but with different tradeoffs involved. To oversimplify, 401(c)3 are

There are several other tax-exemption provisions which do apply to political organizations, but with different tradeoffs involved. To oversimplify, 401(c)3 are exempt from taxation and donations to them can be deducted from the donor's income tax liability. As a tradeoff, they cannot participate in political activity. 401(c)4 organizations are permitted to participate in politics as part (but not all) of their activity, and are exempt from taxation, but donors to them may not deduct their donations. 527 organizations exist explicitly for political purposes and are exempt from taxation. However, donors may not deduct their donations, and 527 organizations are subject to more rigorous regulation and reporting requirements due to their almost guaranteed exposure to campaign finance regulation. The best way to think about it is this: while the IRS permits many types of non-profit organizations to not pay corporate income tax, the ability of donors to deduct donations from their personal income tax is a privilege specifically granted to charities. Charities are a subset of nonprofits which exist for specific purposes, and political advocacy is not one of them. Subject to state law, it is often possible to incorporate nearly any type of business venture as a nonprofit (even a conventionally for-profit one, such as a retailer). However, the IRS does not grant tax-exempt status to any nonprofit. Only nonprofits of specific types are granted tax-exempt status.