Sure, there's absolutely such a thing as an alpha male. But, it's important to distinguish between being AN alpha male, and being THE alpha male. It's a subtle, but important distinction.
Sociologically, every social group that includes more than one male has an alpha male. A guy can be the alpha in his social circle for any number of reasons. He doesn't have to be the biggest, strongest or best looking guy in the group. Sometimes he might be the smartest or the most charismatic, and sometimes there may be no particular reason. And while a guy might be the alpha in his social group, when he's inserted into a different social group with an already established social hierarchy that guy may not be the alpha.
I personally have several distinct social groups that I find myself in from day to day. In some of those social groups I think I'm pretty clearly the alpha, especially at work. I don't day that out of an inflated sense of ego or a need to be important. Simple observation can tell you if you're the alpha. When there's a decision to be made as or for the group, does your opinion seem to carry more weight than that of others? Will people gladly defer and allow you to make the decision for the group? Are you the social catalyst for the group? Does the groups social activity revolve around your schedule and social preferences? All basic clues, but honestly, it's usual very clear to everyone in the group whom the alpha is , even if it's not something that anyone in the group is actively thinking about.
But, I also run in several distinct social circles that where I'm clearly not the alpha. It's a fluid thing. All social groups function in this way, including female groups. Either you are the alpha in a particular group or you aren't. It's something that is sensed very innately by every member of the group almost immediately, without needing to be verbalized.
Now, there's also the very rare case of the guy that is always the alpha male in every social circle he finds himself in. These guys are exceedingly rare, if for no other reason than the fact that social dynamics can fluctuate greatly with cultural differences and education levels. Any of us may have met one or 2 people in our lives that can insert themselves in any social group and be clearly identified as the alpha by all members of the group. Guys that meet this criteria can be called AN alpha male. But that doesn't mean you're not THE alpha male in your social circle.