Private Taxis (個人タクシー) are taxis owned and driven by one person, in opposite to corporate owned taxis, which are many taxis owned by one company.
It can be quite confusing for someone who can’t read Japanese to see all these taxis with the same sign 【個人タクシー】 (Private Taxi) on top of the car, without being from the same company. But while they all have the same sign, they are all privately owned by different drivers. Some also have 【個人市タクシー】 (Private Owned Town Taxi) on top, which one might confound for taxis owned by the town, while they actually aren’t. Taxis with the “市”, or town kanji in there name are also privately owned, just part of a different union.
Private Taxis and corporate taxis are usually in the same price range. The only difference being that corporate taxis often offer members cards to get points, which can be exchange for small presents or fare money.
Another difference is the service. Because private taxi drivers are their own boss, some can end up giving a lesser, or even a bad experience to the passengers. On the other hand, if you had any kind of problem with a corporate taxi, you can easily call their call center to explain what happened, which might lead to repercussions to the driver in question, scare that private owners don’t have to live with.
However, that doesn’t mean that private taxi owners are incompetent. The driver must work in a taxi company for at least 10 years in order to get a private taxi license.