Revisiting the definition of identification.
Although your definition of consistency is fine, I think you're defining identification in a somewhat odd way, especially with regards to the usage of "if we have enough data."
Although we can loosely think about identification as having "infinite data," I'd suggest you instead consider a scenario where you know the true distribution of the observed data.
In this sense, let $P$ denote the true distribution of observed data where $P \in \mathcal{P} \equiv \{P_{\theta} : \theta \in \Theta\}$. We are interested in $\theta$ or some function $f(\theta)$.
Since $P \in \mathcal{P}$, we know that there exists some $\theta \in \Theta$ such that $P = P_{\theta}$. However, given $P$, we cannot distinguish $\theta$ from any other $\theta'$ such that $P = P_{\theta'}$. In words, this is saying that given $P$, we may not 'know' enough about $\theta$ to uniquely pin it down.
To illustrate when this can happen, suppose we observed $P = N(a+b,\sigma^2)$, so that $\theta = (a,b,\sigma^2)$, and we are interested in $f(\theta) = (a,b)$. Given $P$, I cannot uniquely pin down $f(\theta)$, because even though I know $a+b$, I can choose any $f(\theta) = (\theta_1,\theta_2)$ such that $\theta_1 + \theta_2 = a+b$. To make things super concrete, suppose $a = -1,b=1$ so that $a+b = 0$. Then both $(-1,1)$ and $(0,0)$ are consistent with $P$. Hence, even fully knowing the distribution $P$ does not give me enough to pin down the 'true' value of $(a,b)$.
In general, given $P$ and $\mathcal{P}$, the best we can say about $\theta$ is that $\theta \in \Theta^*(P)$ where
$$\Theta^*(P) \equiv \{\theta \in \Theta : P_\theta = P\}.$$
This is simply defining $\Theta^*(P)$ to be the set of all $\theta$ that agree with the observed distribution $P$. We call this the identified set. We then say that $\theta$ is identified if $\Theta^*(P)$ is a singleton for all $P \in \mathcal{P}$. Here, by singleton, we are saying that given $P$, we can uniquely pin down $\theta$. We similarly define these terms for $f(\theta)$.
Relationship between identification and consistency.
It should hopefully be somewhat clear from the above that identification and consistency are closely related.
In particular, if $\theta$ is not identified, then it follows that consistent estimators cannot exist for $\theta$. Why? Well suppose that we had a consistent estimator $\hat{\theta}$. Because it is consistent, it should converge to all values in $\Theta^*(P)$. Since $\theta$ is not identified, then there are values $\tilde{\theta},\bar{\theta} \in \Theta^*(P)$ such that $\tilde{\theta} \neq \bar{\theta}$, and $\hat{\theta}$ cannot converge to two distinct values!
Conversely, if $\theta$ is identified, then consistent estimators may exist, though do not have to. Though most of the time, there will exist a consistent estimator (by appealing to law of large numbers, continuous mapping theorem, and so on), but exceptions exist (i.e., the mean for Pareto distributions with $\alpha < 1$).
