When talking about the risks of radiation in X-rays, often it is compared to background radiation. However, scans are usually aimed at a concentrated area of the body in very short amount of time, whereas I assume background radiation is spread across the whole body. Can the two be compared when we talk about risk to individual organs? Does 1 mSv aimed at one part of the body does more damage to that part than 1 mSv worth of background radiation?

Biological effects from radiation are categorized into two types: “deterministic” and “stochastic.” Deterministic effects occur at high levels of exposure wherein the severity of effects increases with dose. (Hair loss and skin irritation are two common examples.) The amount of radiation required to produce deterministic effects is beyond the range used in diagnostic imaging, but they are sometimes encountered at the high doses involved in radiation therapy.
With stochastic effects, the risk of effect increases with radiation dose. That is, an effect may or may not occur in a patient but its probability of occurring is assumed to scale with dose. The most common example of a stochastic effect is cancer induction (ie, carcinogenesis) which is known to be associated with radiation exposure based on evidence from groups such as atomic bomb survivors, residents near sites of severe nuclear accidents, etc, who received moderate to high radiation doses.
Not all tissues and organs are equally sensitive to radiation, and some are more likely to undergo DNA changes that can eventually lead to stochastic effects such as cancer. The quantity that accounts for these differences in tissue response is called “effective dose.” It is a sum of the radiation doses received by individual organs and tissues after applying weighting factors that account for the radiation sensitivity of the exposed organs, with more sensitive organs carrying more weight than less sensitive organs. The unit of effective dose is the sievert (Sv) and 1 Sv corresponds to 1 joule/kilogram (1 J/kg) of photon energy absorbed uniformly in the whole body. It is common to report effective doses in milliSieverts (mSv, or 0.001 Sv).
Now, to finally address your specific questions: Effective dose is a useful quantity for roughly comparing the relative risks associated with partial-body exposures (such as from X-ray imaging) and whole-body exposures (such as from natural background radiation). By its definition, 1 mSv of effective dose correlates to the same risk of overall detriment regardless of which parts of the body were actually irradiated.
A couple final comments: For the relatively low dose of 1 mSv being considered the duration of irradiation is not a factor, so it is not unreasonable to compare the effective dose from a brief medical imaging procedure to that from a year of background radiation. It is also important to emphasize that effective dose was originally designed as a metric for use in conservative radiological protection calculations and it should never be used to estimate the absolute risk of stochastic effects from radiation exposure.

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