The reason it's called Batbarian is because you control a barbarian and his sidekick, a magical bat. The game opens with them being chased off a cliff by a horde of ogres. The barbarian falls so long that the game introduces you to its sense of humor by doing a gag based on the guy falling so long that he starts getting bored. Eventually he hits the ground, deep inside a cavern. He picks himself up, reunites with his bat friend, and they head off to figure out how to escape this strange place.
The game is a Metroidvania with the gimmick that the bat is luminous and can be manipulated by tossing different kinds of fruit for it to eat, and you can get power-ups that allow you to switch among different elemental affinities. Up against an ice-based enemy? Switch the bat to fire mode to make the enemy thaw enough for you to damage it with your sword. Some switches are operated by light, so you'll need to toss a fruit over to the switch so the bat's light will turn on the switch and open the door or create the platform you need to advance. It's an interesting concept in a genre that has rapidly become crowded over the last decade.
It's a light-hearted, humorous game, with the humor slanted more toward farce. It's not funny in the same way as Age of Barbarian, where the humor is based on a specific familiarity with the genre, but more like something you'd find in the old Dragon magazine comic strips, broadly poking at fantasy cliches. Sometimes it's genuinely funny, but most of the time it's merely smile-inducing.
It's a pretty nice-looking game. The graphics are a bit on the miniaturized side, but they still convey a lot of personality and atmosphere. On the downside, between the player character, the bat, and the extra sidekicks you can acquire through the course of the game, things can get very chaotic when combat picks up a lot. Are you tracking your guy or your partners along with the bad guys?
It's also probably one of the more challenging examples of its genre. Metroidvanias tend to emphasize exploration more than anything else and often make the player feel very powerful by the end thanks to all the powers you gain, but Batbarian likes to make the player work for everything by throwing out increasingly tricky combinations of enemies and requiring deft handling of the hero, the bat, and your partner. Although there's an RPG-like leveling system, upon a level-up the game presents the player with a slot machine that, based on when you click, will determine if your offense, defense, or "awareness" increase (awareness seems to increase the radius of the bat's illumination?). The slot machine isn't too hard to use, but it's also a needless bother and if you're not paying attention to it, you might get stuck picking up more awareness increases than is advisable and then handicapping yourself in terms of attack/defense power by the later stages of the game.
The biggest issue the game has is that it's yet another pretty good Metroidvania in a genre that's stuffed with similarly pretty good examples. You could pick it up and have a decent time playing it, but you could say that about many other games as well and when really top level games like Hollow Knight are out there, you need to be better than just decent to stand out. The barbarian theme is amusing but not explored very deeply beyond gags about how he expresses himself in dialogue scenes (do you pick the brutish option or the eloquent one?).