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Axial tilt does precess, over a time period of some 40,000 years. In order to have a season lasting multiple years though, the precession would have to have a period of approximately a year. Explanation: Let's say you're living in the northern hemisphere, it's December, the Earth's axis is tilted to the right in space, and the Earth is located on the right of the Sun in space. Therefore it's winter where you are because the northern hemisphere is tilted away from the Sun. Six months later it's June and the Earth is on the left of the Sun in space, and you want it to still be winter. Therefore you need to still be tilted away from the Sun, so now the Earth's axis has to be tilted to the left. Repeat. Such a fast precession is highly unlikely physically, and would probably require something awkward like multiple moons, a single huge moon with an impossibly high density, or another planet's orbit being close to the main planet's.

You're right that precession (the wobble of the tilt) is caused by gravity from another body such as our moon. As for what causes the tilt in the first place, it depends upon the planet. For the Earth, we think it's the same collision with a Mars-sized body that created Earth's Moon. I'm not sure for Venus (177 degrees) or Uranus (97 degrees), but I'm guessing a collision too. Smaller tilts such as Jupiter's 3 degrees probably were just a result of eddies in the cloud from which the planets and Sun all formed.