There is now strong evidence that the metal-rich globular clusters (GCs) near the center of our Galaxy are associated with the Galactic bulge rather than the disk as previously thought. Here we extend the concept of bulge GCs to the GC systems of nearby spiral galaxies. In particular, the kinematic and metallicity properties of the GC systems favor a bulge rather than a disk origin. The number of metal-rich GCs normalized by the bulge luminosity is roughly constant (i.e., bulge SN~1) in nearby spiral galaxies, and this value is similar to that for field elliptical galaxies when only the red (metal-rich) GCs are considered. We argue that the metallicity distributions of GCs in spiral and elliptical galaxies are remarkably similar and that they obey the same correlation of mean GC metallicity with host galaxy mass. We further suggest that the metal-rich GCs in spiral galaxies are the direct analogs of the red GCs seen in elliptical galaxies. The formation of a bulge/spheroidal stellar system is accompanied by the formation of metal-rich GCs. The similarities between GC systems in spiral and elliptical galaxies appear to be greater than the differences.