Is the equinox Sun always at the same zodiac sign in the ecliptic belt? During the year the Sun moves along the ecliptic passing all the signs of the zodiac belt, in my understanding. This means that it takes ~1 month to go from one zodiac sign to the next.  

The equinoxes are approximately on 20th of March and 23rd of September. This means the vernal equinox Sun is in Pisces (or Aries) and the autumnal equinox Sun is in Libra (or Virgo).  

But also I found that there is the observation of precession of the equinoxes, where the equinox Sun passes through all zodiac signs in a calendar round that lasts ~25,000 years.  

How is this possible?   

If my understanding about the Sun's yearly round across the zodiac is correct, the equinox Sun should always be at the same zodiac signs. How can this precession happen?
As you said, the Sun takes a year to cycle the Zodiac, and it takes a year between two spring or autumn equinoxes. Both statements are true, but "year" in them means two different things. It takes a sidereal year to cycle the Zodiac, but it takes a tropic year to cycle between equinoxes. Since the tropic year is about 20 minutes shorter, the equinoctial points move across the Zodiac cycling it in about 25,000 years.

The calendar year we use in the Gregorian calendar is designed to approximate the tropic year to have it synchronised with seasons.

The reason for all it is that the Earth's axis is not fixed but its orientation slowly rotates.