Over the years of superheroes not many superheroes' symbols have changed as much as Batman (my opinion, not fact).
Why hasn't Batman had any concrete symbol?
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The simplest answer is Batman has had a lot of different interpretations. He is one of DC Comics most highly interpreted and recreated characters. Those images are not the only representations, the Bat-logo on his chest has an equally impressive visual pedigree.
Each of those symbols is a different property attempting to bring a unique perspective to the Dark Knight. Each of them was seeking an icon while reminiscent of the Batlogo on Batman's chest (which has also had lots of permutations) was still unique and distinctive enough to represent THEIR take on the Batman.
When you consider how many different Bat properties there are, the number of logos makes sense:
While Batman has a wide array of image icons used for media and marketing, I suspect you will find almost every well marketed character will have a similar bevy of icons used to represent them.
Batman is unique only because he has such a strong symbol that has ignited the imaginations of artists everywhere to take that symbol and make memorable and yet recognizable versions of that very first and very simple logo.
There are several possibilities. The first one, and perhaps the strongest, is that many of those symbols come from different universes. Each Animated Series (excepting Batman: The Animated Series / Batman Beyond / Justice League) has it's own universe and continuity. The bat symbols within those universes are typically consistent. Batman's logo is much more complex than most other superheroes' logos. It's a bat silhouette, but there are many types of bats in the world, and many points on the realistic/iconic scale at which Batman could stop. Each universe has a radically different art style, and correspondingly different bat logo.
The other possibility that springs to mind
Batman is unique among DC superheroes. Batman, alone amongst all of the top tiers, is completely lacking in super powers.
He fights crime and evil with ingenuity and training, and a boatload of gadgets. Bad guys eventually acclimate to any given stimulus, as any human does. Enough times seeing Batman, people will stop having as much fear of him. And without inspiring fear in his foes, Batman's job gets a lot harder.
In response, he changes things. His appearance changes, and it is fresh - people have the same gut reaction as they did the first time they saw him. It's a tiny thing, but it gives him just that much more of an edge.
Additionally, Batman adjusts his gear based on his experiences and his opponents. When he was being pursued by the police in Frank Miller's Batman: Year One, he had a heavily armored torso. Thus, he put the logo there, with a yellow circle. Cops, who practice regularly with targets and who are trained to aim for the center of mass, shot him in the chest. It bruised, but that's all - Batman put the target on his chest specifically for that reason.
Other suits are designed to make him harder to see and give him more freedom of movement - they aren't as heavily armored and he relies more on stealth. This calls for a change in the logo, as bright colors are contraindicated. Larger, sweeping designs are better for blending in with shadows, so he uses those.
Green Lantern? Flash? Superman? Wonder Woman? They don't have to deal with that. A simple man with a 9mm is never going to be a threat to them. Batman is mortal and frail, he takes every edge he can get.
It seems that, differently from another characters' symbols, the classical bat logo is remarkable not by a specific design (such as the Green Lantern's symbol), but for the idea behind it. So it is more prone to be changed over time. By the way, not only his symbol, but also his uniform has different versions.