"Making adequate identification assumptions is sufficient for identifying causal relationships" is either tautologically true or obviously wrong. It is true if by "adequate identification assumptions" you mean "assumptions that identify a causal effect".
If you mean "adequate" in the sense of "substantively adequate", then of course making such identification assumptions does not always guarantee that you can identify a causal effect. This is what most discussions in the social sciences are about: People question the validity of identification assumptions, state that they should be weakened, and then argue that for this reason, the effect is not identified.
To give a short definition of causal identification, it means you can write a causal parameter in terms of the probability distribution of the observable variables.
E.g., when you think the causal effect of $X$ on $Y$ is a constant $\beta$, one may identify it with $cov(X, Y)/(var(X))$, the coefficient of a linear regression of Y on X, assuming $E[\epsilon|X]=0$, where $\epsilon$ is the structural error representing all other causes of Y other than X.
Or, when you define the causal parameter in potential outcomes, for example as the ATE $E[Y^{1} - Y^{0}]$, then you sometimes can identify it as $E[Y|X = 1] - E[Y|X = 0]$, assuming $E[Y^{x}|X] = E[Y^{x}]$ for $x \in {1,2}$. The latter assumption is a generalization of the error term assumption: $X$ should be mean-independent from all variables that affect $Y$ save for $X$.
