A “wobbler” isn't a misdemeanor, nor is it a felony. It's both. It's a crime that prosecutors can charge as and judges can sentence as either a misdemeanor or felony.
Consider, for example, California's statute regarding assault with a deadly weapon. That statute provides that an assault with such a weapon (other than a gun) can be punished by either prison time of two, three, or four years, or up to 364 days in jail. (Cal. Penal Code § 245(a)(1) (2019), Cal. Penal Code § 18.5(a) (2019).) Prison time denotes a felony, while jail time indicates a misdemeanor. If the judge sentences the defendant to prison time, a felony conviction results. If the judge chooses jail time or a lesser punishment, it's a misdemeanor conviction. (Some states even allow judges to wait to decide whether certain crimes are misdemeanors or felonies until the defendant has completed probation.)
Prosecutors have the choice of whether to charge a wobbler as a felony or a misdemeanor. But even when prosecutors choose one over the other, judges typically have the final say. For example, if prosecutors charge a wobbler as a felony, the judge can reduce it to a misdemeanor at sentencing.
The facts of the crime and the circumstances surrounding the offender often dictate whether prosecutors and judges go with the misdemeanor or felony label. Stabbing someone with a knife, for example, is a kind of deadly-weapon assault likely to incur a felony charge or conviction. On the other hand, swinging a wine bottle in the general direction of someone might result in a misdemeanor, particularly if the judge determines that prison time wouldn't have a rehabilitative effect.