Is Time Really the Fourth Dimension?

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How about summarizing the video and relating how you understand what it is saying.
The term "the fourth dimension" has a variety of meaning depending on who you are talking to.

In relativity, time can be treated as a special case of a dimension of space.
There are models which include many more dimensions. Michio Kaku is relating the kinds of ideas that lead to these in this video. Time would certainly be included in any multi-dimensional model ... though, off Kaku's description, time would have to be the 12th dimension since he describes 3D universe-bubbles on the surface of an 11D multiverse which are expanding in time (implying that the 11D mess is evolving in time).

I'm not sure that's the impression he intends - we have to be careful not to read too much into these kinds of lay-descriptions: they are all hopelessly lacking.
In this case Kaku is trying to answer a question - and the questoner appears to have left time off the number of dimensions that our universe has.
 
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Let's put it this way. It takes 3 separate numbers to describe the spatial position of an object relative to a point in space or another object. Since it takes 3 numbers, we refer to each of these as being a "dimension" of space. If we want to describe the position of something in space AND time it takes 4 numbers, with time being the 4th. (It's a little more complicated, but I trust you get the general idea)
 
Yeah - number of dimensions could just be seen as the number of degrees of freedom.
eg. A bicycle can be described by numbers:

x,y,z location (3)
orientation (lean, yaw and pitch) at the location (3)
the relative rotation-angle of each wheel (2)
the relative turn-angle of the handle-bars (1)
the angle of the pedals (1)
time it got there (1)
height and rotation of the seat (2)
... that's 13 dimensions, and I have not gone into it's gears yet.
 

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