causal vs statistical identifiability

[d-morrison]

can causal identifiability be represented as statistical identifiability?
for example, in DiD, we assume there is no group * time interaction term (since it would be identical to the exposure term); but if we put them both in the model, we introduce collinearity, which causes statistical nonidentifiability.

Can we always do this? I think yes; we can always create two indicators, "treated group" and "treatment", where "treated group" represents uncontrolled confounding