Without trying to precisely define scientific methodology, I think it always needs some connection to empirical evidence. This is the problem with the Drake equation, but also with some other speculative fields like string theory.
So I'd say that the Drake equation is scientific in the sense that it's motivated by scientific speculation, but has not yet found a form that can be tested.
In terms of renown, it is probably it the top equations people would be able to recognize by name, and tell you what it is about. The only equations that might be more recognizable by name are the Pythagorean theorem, ideal gas law, and perhaps some Newtons laws of motion.
In terms of scientific relevance, It is a tool for examining what many people consider the single most important questions ever asked.
If it’s a thought experiment, stop calling it an “equation”. People hear that something is an equation and they think that there’s real math and science involved. Call it the “Drake Conjecture” or something.
To this day, as WP notes, 'It is typically not possible to solve the Schrödinger equation exactly for situations of physical interest.' Yet he got a Nobel prize for it. Frank used his equation to get us thinking about SETI, and SETI is now decades old.
Yes, just like there are people who consider Crystal Power serious science. A lot of people are deeply impressed by the Drake Equation, presumably because they can't understand it.
Uh, this might be a tad bit hyperbolic. Does anyone even consider it serious science?