Why isn't this a problem with heritability studies? In many studies of heritability, researchers study identical twins raised apart and raised together, and see how much a trait correlates between the two types of twins. Sometimes they use fraternal twins, there are a few variations on the scheme.
But while we can measure how similar two people are in terms of their genes (at least, we can do this pretty well) we have no good measures of how much the environment varies. Human environments vary tremendously, of course. They vary over both time and place, in ways that are impossible to measure. 
This leads to three sorts of problems:
1) If identical twins are assumed to have identical genes, then any variation is due to non-genetic factors. What factors are there that are not environment or genes?
2) If we assume that fraternal twins have 50% genes in common, then we can say they have half as much in common as identical twins. But if we put one twin in a farmhouse in Iowa and another in a farmhouse in the next county over in Iowa, then environment won't vary much; on the other hand, if one twin is put in a farmhouse in Iowa and the other in an apartment in New York City or Lagos .... well, that's a lot of variation. But how much? 
It seems to me there's no way to turn this into a number, and so it seems to me that heritability coefficients are a kludge and inherently silly.
There is a third problem, which @henrik 's comment brought back to mind: Genes and environment can interact. This makes for even more problems, because, as we all know, estimating main effects in the presence of interaction is problematic. 
For an extreme case, take the problem of phenylketonuria. Mortality in this disease is entirely due to an interaction between genetics and environment. Less extreme cases abound. 
But a lot of smart people use them, so maybe I am missing something.
 A: I totally agree with your idea that the question of the environment is critical in designing a twin or similar study. However, I think you miss one important point from the classical twin studies which compare monozygotic and dizygotic twins. Wikipedia gives a nice overview which I cite next: 

The power of twin designs arises from the fact that twins may be
  either monozygotic (MZ: developing from a single fertilized egg and
  therefore sharing all of their alleles) – or dizygotic (DZ: developing
  from two fertilized eggs and therefore sharing on average 50% of their
  polymorphic alleles, the same level of genetic similarity as found in
  non-twin siblings). These known differences in genetic similarity,
  together with a testable assumption of equal environments for MZ and
  DZ twins (Bouchard & Propping, 1993) creates the basis for the twin
  design for exploring the effects of genetic and environmental variance
  on a phenotype (Neale & Cardon, 1992). [...]
Like all behavior genetic research, the classic twin study begins from
  assessing the variance of a behavior (called a phenotype by
  geneticists) in a large group, and attempts to estimate how much of
  this is due to genetic effects (heritability), and how much appears to
  be due to shared or unique environmental effects - events that affect
  each twin in a different way, or events that occur to one twin but not
  another.
Typically these three components are called A (additive genetics) C
  (common environment) and E (unique environment); the so-called ACE
  Model. [...] Given the ACE model, researchers can determine what
  proportion of variance in a trait is heritable, versus the proportions
  which are due to shared environment or unshared environment. 
Monozygotic (MZ) twins raised in a family share both 100% of their
  genes, and all of the shared environment. Any differences arising
  between them in these circumstances are random (unique). The
  correlation we observe between MZ twins provides an estimate of A + C
  . Dizygous (DZ) twins have a common shared environment, and share on
  average 50% of their genes: so the correlation between DZ twins is a
  direct estimate of ½A + C . If r is the correlation observed for a
  particular trait, then:
rmz = A + C

rdz = ½A + C

Where rmz and rdz are simply the correlations of the trait in MZ and
  DZ twins respectively.
Twice difference between MZ and DZ twins gives us A: the additive
  genetic effect (Falconer's formula). C is simply the MZ correlation
  minus our estimate of A. The random (unique) factor E is estimated
  directly by how much the MZ twin correlation deviates from 1. (Jinks &
  Fulker, 1970; Plomin, DeFries , McClearn, & McGuffin, 2001).
It can be seen from the modelling above, that the main assumption of
  the twin study is that of equal environments. At an intuitive level,
  this seems reasonable – why would parents note that two children
  shared their hair and eye color, and then contrive to make their IQs
  identical? Indeed, how could they?

As described, the classical paradigm does not ignore the question of environment but assumes that MZ and DZ twins share an equal amount of environment. A seemingly reasonable assumptions. However, I totally agree that it is debatable.
Therefore, in the modern debate on heritability, twin studies of this sort alone are not sufficient. There are other types of designs and relationships that need to provide convergent evidence for claims, most notably adoptions studies. In these studies you compare the correlations between (e.g.,) adopted children and their biological parents, their adoptive parents, their biological brothers and sisters and adopted brothers and sisters. When taking all these correlations into account it is quite easier to partition the ACE effects as you have more different relations (gene wise and environment wise).
The best would be to also have some twins in these studies :-)
