In econometrics, identification can take several forms depending on the type of model you are working with (see this survey for a more comprehensive description). However, in general, identification pertains to the idea that either your econometric model or the exogenous variation in your data (or both) allows you to tease out the parameters that truly describe the data generating process.
To be more specific, suppose that you are studying outcome $y$ which depends (stochastically) on $x$ so that the data is generated according to $F(y|x)$ where $F$ is some distribution function. Then you have an econometric model that depends on a parameter, $\theta$ (this can be a vector of parameters), and for each parameter, it spells out a conditional distribution of $y$ in terms of $x$. That is your econometric model is a function $M(\theta)=G(y|x; \theta)$. We say that the model is identified if there exists a unique parameter, $\theta^*$, in the domain of $M$ such that $M(\theta^*)=F(y|x)$.
In reduced-form work, $M$ is assumed to be linear (or in general to be an injective function and have a relatively simple form), so ensuring identification has more to do with your data, $F$, having the right properties (mainly having enough exogenous variation).
In contrast, structural work typically deals with data that is not experimental or quasi-experimental so $M$ cannot credibly be assumed linear. Instead, you develop a rich enough model and identification requires $M$ to be invertible. If $M$ is invertible, then you have that $M^{-1}(F(y|x))=\theta^*$.
I think that in relation with identification, linear algebra tools are mainly useful to prove the invertibility of $M$. However, it can prove also useful when it comes to designing an estimation strategy, and when implementing the estimation strategy in the computer, etc.
