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I finally found a solution. I post it just in case it becomes useful for somebody else in the future: apparently the problem is related to how astropy deals with stellar aberration correction. For near-Earth objects this correction should not be included, and this can be achieved using a topocentric ITRS frame. Following this link, the steps to follow are:

Obtain a cartesian representation of the object and the observatory in the ITRS frame
Subtract the former two vectors (i.e.obtain the relative position of the object with respect to the observatory).
Pass this relative coordinates to ITRS, this provides the topocentric ITRS location of the object.
Transform from topocentric ITRS to AltAz.

Code:

#Problem calculating distance to object in the sky
#worse when close to observer. Negligible if distance > 100 km
from astropy.time import Time
from astropy import units as u
from astropy.coordinates import EarthLocation, AltAz, ITRS
obs_lon = 0*u.deg
obs_lat = 0*u.deg
obs_alt = 0*u.m
aircraft_lon = 0*u.deg
aircraft_lat = 0*u.deg
aircraft_alt = 1000*u.m
obs_time = Time ('2000-01-01T00:00:00.000')
obs_location = EarthLocation.from_geodetic (obs_lon,obs_lat, obs_alt)
aircraft_location = EarthLocation.from_geodetic(aircraft_lon, aircraft_lat, \
                                                aircraft_alt)
aircraft_altaz_old = aircraft_location.get_gcrs(obstime=obs_time). \
    transform_to(AltAz(location=obs_location, obstime=obs_time))
dist = ((aircraft_location.x - obs_location.x)**2 + \
        (aircraft_location.y - obs_location.y)**2 + \
            (aircraft_location.z - obs_location.z)**2)**0.5
#New block of code solving the issue:
aircraft_itrs = aircraft_location.get_itrs(obstime = obs_time)
obs_itrs = obs_location.get_itrs(obstime = obs_time)
aircraft_itrs_relative = aircraft_itrs.cartesian.without_differentials() - \
    obs_itrs.cartesian
aircraft_topo = ITRS(aircraft_itrs_relative, obstime = obs_time, \
                     location = obs_location)
aircraft_altaz = aircraft_topo.transform_to(AltAz(obstime = obs_time, \
                                                  location=obs_location))
print("dist calculated from xyz", dist) # this gives 1000 m, as expected
print("dist returned from AltAz (old)", aircraft_altaz_old.distance) # 1189.6 m
print("dist returned from AltAz (new)", aircraft_altaz.distance) # 10000 m
print("az, alt ",aircraft_altaz.az.to_value(), aircraft_altaz.alt.to_value())
#now distance, az and alt are OK

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