The constitutional limitation is on “bills for raising revenue,” and that is laws that impose a tax. The courts construe that very narrowly, so that fees are not considered bills for raising revenue. The debt limit is not a tax, at all. Spending is not a tax either.
The debt limit still have to be approved by both houses and signed by the President - at least to my recollection. Neither the President nor the Senate has the power to raise that by themselves.
Am I wrong here?