A common counter-apologetics argument, this applies to almost every argument for the existence of God other than those based on a specific scripture or revelation. These arguments generally attempt to show the existence of a being with some particular trait, then commit an equivocation fallacy wherein the being in question is labelled "God" and presumed to be an intelligent, supernatural, and usually monotheistic deity (along with possessing any other traits the apologist wants to believe in). In reality, nothing like "God" has usually been proven. At best, the argument applies to the existence of some completely unknown entity with one specific trait.
Many arguments do not demonstrate anything like a God at all
The fine-tuning argument could easily be used to demonstrate the existence of universe-creating extraterrestrials (who may or may not have any other traits attributed to God). Similarly, the natural law argument could be used to demonstrate that the universe was designed by a congressional committee. The first cause argument doesn't even require an intelligent cause (in fact, it might make more sense for a first cause to be unintelligent, because otherwise one could ask "Why would this thing with no cause or design behind it at all have such a complex trait as the ability to think?").
Many arguments for the existence of God fail to distinguish between different gods. If an argument such as the Kalam Cosmological Argument fails to distinguish between the Muslim conception of Allah and the Hellenistic conception of Chaos, that should make it clear how weak the conclusion of the argument really is. Far from being an earth-shaking discovery, the argument is effectively a fancy way to say "something made all this junk", without stating what that something is or how that knowledge could possibly be useful or predictive of anything.
Not only can most of these arguments be used to support most modern religions, they also support "dead" religions, as well as many potential future religions or religions that might never even be thought up. The ancient Egyptians could not have become Christians. Similarly, we cannot be members of a religion that doesn't currently exist. That doesn't, however, prove that such a religion could not be true.
The only reasonable, honest solution to this problem is to not accept proof of any religion unless that proof is specific to that religion's claims. This also applies to arguments such as Pascal's Wager.
Different arguments might address different Gods
Let's pretend that one can prove the existence of a first cause, a designer of our universe, a designer for life, an architect of objective morality, a god who runs the afterlife, and any number of other things. It's not clear why one should assume that these are all the same entity. In fact, polytheistic religions explicitly invoke a number of gods running different aspects of the universe. One cannot demonstrate the existence of a single monotheistic God that does all of these things without going through them individually and showing why all these different roles go together.