No, that's not exactly a sci-fi concoction. In special and general relativity, there are three dimensions for space and one dimension for time, and this is not something that is of "incidental" importance to special / general relativity, it's a pretty essential shift in perspective to these theories to think of the universe as (curved) four-dimensional spacetime.
But "dimension" is something mathematical. I would say it doesn't quite make sense to say "is the fourth dimension time" in the same way as it wouldn't make sense to say "is the fifth an apple?" The same way that numbers can refer to different things in different contexts (including in the context of different scientific theories), dimensions can correspond to different things in different contexts. For example, statistics and machine learning heavily use "high dimensional" mathematics, but there the "dimensions" would correspond to different variables you are trying to predict or explain. E.g. if you were trying to predict chance of heart attack from 1000 different factors, then you would have 1000+1 total "dimensions," and in that case the "fourth dimension" might be "cigarettes smoked per week" (rather than time).
Contextually of dimension even exists within a specific scientific theory. In relativity, the direction you call time might contain some component of the direction I call space. This implies notions like simultaneity are not well defined in a universal context.
No, 4D spacetime is a real thing in physics, which explains things like time dilation and the speed of light. But sci-fi does tend to abuse the term "dimension" for other ideas that are not scientific.
yeah, nobody can visualize it. it's something you just get used to after a while.
there's an old joke about a mathematician teaching an engineer about thirteen-dimensional spaces. "What do you think," the mathematician asks. "My head's spinning," the engineer confesses. "How can you develop any intuition for thirteen-dimensional space?"
"Well, it's not so hard. All I do is visualize the situation in arbitrary N-dimensional space and then set N = 13."
But "dimension" is something mathematical. I would say it doesn't quite make sense to say "is the fourth dimension time" in the same way as it wouldn't make sense to say "is the fifth an apple?" The same way that numbers can refer to different things in different contexts (including in the context of different scientific theories), dimensions can correspond to different things in different contexts. For example, statistics and machine learning heavily use "high dimensional" mathematics, but there the "dimensions" would correspond to different variables you are trying to predict or explain. E.g. if you were trying to predict chance of heart attack from 1000 different factors, then you would have 1000+1 total "dimensions," and in that case the "fourth dimension" might be "cigarettes smoked per week" (rather than time).