In its current form, manzai only dates to the 1920s, when all elements other than comic dialogue were removed.
That would be very close to it. As a matter of fact, in the link Omi no Kami provided, the Two Beats start talking about baseball (hence the swinging gestures) in a way that’s very reminiscent of Who’s on First?
Also worthy of mention is rakugo, while technically it’s sit-down rather than stand-up comedy, unlike manzai it features only a single performer. Here’s a good example. In its current form, it can be traced back to the 17th century, making it technically older than manzai and much older than American stand-up. Here’s another example, in English (it’s not half as funny as in Japanese as much of the humour lies in the choice of words). While a lot of rakugo is based on a set of standard stories, which act like jazz standards, there’s also some stuff that’s very close to American stand-up monologues.
Somewhat half-way between rakugo and manzai, is mandan which appeared in the 1930s. Here’s one of the best mandan performers, Kimimaro doing his stuff. Content-wise, it’s also very, very close to stand-up.
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