Both terms are fairly loaded terms with meanings that will depend on who is using it.
Broadly speaking, "empirical strategy" is an umbrella term used by researchers to indicate their overall "process" in approaching a question and delivering an answer. Indeed, Angrist and Krueger write in Empirical Strategies in Labor Economics (my own highlighting):
We use the term empirical strategy broadly, beginning with the
statement of a causal question, and extending to identification
strategies and econometric methods, selection of data sources,
measurement issues, and sensitivity tests.
This re-affirms the idea that empirical strategy is a catch-all term to indicate your overall method.
In contrast, I would argue "identification strategy" means something very specific. I wrote a bit about identification in this CV question. Based on my definition of identification presented there, I would define identification strategy as the process of
- defining a parameter you are interested in (such as the causal effect of treatment on outcome), and
- proving that your observed data (a DGP) and imposed assumptions (such as parallel trends in diff in diffs) identify this parameter
In applied work, you'll probably notice people are not so rigorous about this, and instead will say something hand-wavy such as "our identification strategy leverages a difference in difference design..." This is fine, because in many commonly used designs, previous work has already done the tedious work of showing the formalities, so applied research can simply state that they are doing a diff in diff without having to explain every detail. But in works where the identification strategy is novel, then they have to actually prove the identification strategy "works".