What's the precision of a quartz crystal?
Frequency tolerance vs Frequency stability...

Tolerance involves mostly the making of a crystal blank: grinding, and plating its surface so that its static primary resonance hits the target frequency. Tolerance is spec'd at a fixed temperature, because temperature mostly affects resonance stability. Tolerance of primary resonant frequency is the least-complex spec. Tolerance might include ageing. Or how a crystal ages might be a separate spec (also done at a fixed temperature).

Stability involves mostly temperature. Temperature stability is complicated by rate-of-change of temperature - fast changes of temperature can change resonant frequency radically if one part of the crystal cools/heats more than another part. The crystal spec sheet usually addresses a very slowly-scanned temperature range where the crystal-under-test is uniformly heated. Fast-changing temperature effects are affected by how a crystal is mounted - something over which a crystal manufacturer has no control.

Crystals are usually cut at specific angles where temperature coefficients are near zero. But they remain close-to-zero over a very small range of temperatures. An AT-cut crystal extends this range somewhat because temperature profiles follow a 3rd-order curve. Over the specified temperature range, the derivative of this curve changes sign at two different temperatures. Changing the cut angle very slightly allows a manufacturer to tailor the stability of resonance to a different temperature range. Other crystal cuts follow a 2nd-order temperature curve, where perfect temperature/frequency stability occurs at only one temperature.

A crystal manufacturer can only specify a crystal's primary resonance, preferably its series resonance: the frequency where its mass and spring-constant match. But many crystal oscillating circuits also involve the circuit + crystal capacitance, which shifts the crystal's series resonance to a higher frequency. To accommodate circuit-builders, a manufacturer may specify a crystals' parallel resonance, and include a capacitance in this spec. Ideally, this added capacitance is not affected by temperature. 

You seem to be mostly concerned with frequency stability...measured over a long time-span. A frequency counter or a clock, watch is an example. An oscillator's jitter is not a concern in such an application.

By varying an oscillator's capacitance, its oscillating frequency can be changed over a small range. Doing so usually affects frequency stability detrimentally, partly because the added capacitance isn't temperature-stable, and partly because sharpness of resonance is compromised when shifted from series-resonance. A crystal manufacturer's spec sheet cannot address this added complication.A manufacturer has the difficulty of specifying a crystal that might be used in a few different application circuits (mostly oscillators) of widely-varying topologies. A casual oscillator design will very likely result in frequency variations poorer than the raw crystal spec.
TLDR:


  If I take two consecutive measurements of the same crystal assuming
  constant temperature. Can I expect the frequency error to be the same
  on both measurements (or at least lower than the Frequency Tolerance)?
  Or should both measured errors be assumed independent?


I'm assuming that you're measuring frequency of an oscillator using a single crystal to set its frequency, and you're using exactly the same measurement setup for both measurements, using a frequency counter having much better specs than the crystal-under-test. Also assuming that the whole test oscillator operates at the "constant temperature":
About the only crystal spec that can vary is its ageing spec.

Should you put the same crystal into a different oscillator for the 2nd measurement, expect frequency tolerance of the oscillator frequency to be greater than tolerance of the raw crystal. A crystal driven with more current shifts in frequency too...rarely spec'd in a crystal data sheet.
The components of an oscillator likely vary, for example its DC supply voltage. And you didn't mention a varying load - that will pull frequency +/-... so a 2nd frequency counter (having excellent specs) might measure a different frequency than the 1st. The components of an oscillator might have their own ageing effects. A freshly-built oscillator often drifts more while very young, and improves with age. A long rest-period between two frequency measurements is affected by both crystal ageing, and oscillator ageing.