No, not really.
That doesn't mean there is no training for product management. Product management courses, programs, certifications, and manuals abound. In later posts, I will cover some of them. However, no equivalent of law school or medical school exists in this multi-faceted discipline. To be fair, a formal education is not a prerequisite for most areas of business. There are few (if any?) equivalents of the legal bar or the medical boards for any role in the software industry. Consider that Stanford may be proudest of its dropouts Larry Page and Sergey Brin and Harvard may revel more in the successes of dropouts Bill Gates and Mark Zuckerberg than it does in most of its actual graduates.
So, school won't take you all the way. But what will? The trouble is that product management sits at the center of a Venn diagram with three overlapping disciplines: business, technology, and design. Master one of those and you still aren't a product manager. But become proficient in two? You're getting closer. Being a designer, an engineer, a management consultant could help. Having domain expertise helps. For instance, being a doctor yourself might help you create better medical record software. Same for a lawyer aspiring to create legal products. Any of the aforementioned backgrounds may help, but all of them aren't necessary and each of them won't be sufficient. Actually, being somewhat of a jack of all trades and master of none may be the best course. Just recruit some masters of each of the three disciplines onto your team.
So, alas, no school will perfectly set you up. You will have to figure out what matters as you go and learn it as you go. And never stop learning, because interdisciplinary fields change fast. After all, it was Steve Jobs who said, "The over-all point is that new technology will not necessarily replace old technology, but it will date it. By definition. Eventually, it will replace it."