Songwriting can be taught, yes. In most music courses you start by analyzing the Bach Chorales, which (along with some Gregorian work from the middle ages) is what really kicked off contemporary music composition. By analyzing the Chorales and moving forward from there you learn how to manipulate chord progressions, harmony, point and counterpoint.
Composers classically trained this way tend (!) to have an easier time writing melodies, harmonies, and progressions in a consistent manner, ie not having to wait for "inspiration to strike". The composer, of course, still needs to develop an emotional connection in the music, but the point is that it can, and routinely is, taught.
My girlfriend is a trained classical singer whereas I'm a self taught musician. She doesn't really gravitate towards the rock music I like to write but because of her training she can easily jam, riff, or write anything far quicker than I can. Songwriting is a very technical skill indeed.
Composers classically trained this way tend (!) to have an easier time writing melodies, harmonies, and progressions in a consistent manner, ie not having to wait for "inspiration to strike". The composer, of course, still needs to develop an emotional connection in the music, but the point is that it can, and routinely is, taught.