Not all useful tools for thinking must be considered science. I think it's quite fair to say the Drake equation isn't "science".
However, I think Popper's "conjecture and refutation" is a more interesting model for thinking about the increase of knowledge than falsifiability is.
Through that lens, you might say the Drake equation was a conjecture intended to stimulate refutations, which might lead to further productive theories.
Popper's essays on Parmenides are really interesting on this. Popper frames Parmenides' philosophy as essentially a conjecture that "change is an illusion", and the history of western science since then as various "research programmes" in response to that. I'm summarising terribly, but I think he illustrates well the value of conjectures that aren't necessarily "scientific" in stimulating thought.
The Drake equation is falsifiable. All you have to do is plug the values (which are all things that could technically be measured) into the equation, then count the number of civilizations in the Milky Way, and see if that's equal to what the equation predicted. it is possible that it's not, so the equation is falsifiable.
That it's impractical for us to actually do the experiment just means that we cannot perform the experiment and so it doesn't help us in practical terms. But it is a thing that could be done, strictly speaking.
That said, it's also a thought experiment, not a scientific postulate, and isn't meant to be "scientific".
Certainly useful for thinking, but its lack of falsifiability, predictive power or validation is indeed why there's so much controversy around it and why it remains a hypothesised model rather than widely accepted scientific fact. The best you can do is test the assumptions of the model itself, or its constituent terms.
