There are two aspects to trainer certification and accredition. One is "what exists today", and the second is "what should exist". The first is fairly straightforward, while the second is controversial within the training industry.
First, what is. There is no universally accepted trainer certification or accreditation. In other words, there is no certificate that is universally recognized within the training industry, and there is no process that is recognized and understood by the consumers and purchasers of training.
This is both positive and negative depending on where one sits. On one hand there are no paper qualifications that serve as barriers to people who want to become trainers. On the other hand there are no qualifications that serve to filter out people who want to become trainers who are incompetent or not suited to the field.
While there is a professional association in the U.S., that is THE professional association for trainers (American Society For Training and Development - ASTD) and they offer a certification process, most consumers aren't aware of them, and many trainers put little credence and value in their certification process.
Second, what should be. The discussion of whether trainers should be certified, or licenced is a topic that's been hot for many years, with people weighing in from all sides of the training profession. There is simply no agreement, about whether there should be certification, or how it should be set up, and how to educate consumers about certification.
This is typical of a field that is immature (young) and fractured (many paths into the field). The training field attracts lots of people from other fields, who do not want, or think they need a formal process to learn how to design and deliver training, or any certification. It's not likely to change in the forseeable future.