Can I say something like,
The movie became famous in the 2015s
I want to use it as an approximation; it's not really 2015 but somewhere around that period.
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No, you can’t say “the 2015s”.
You could instead say “around 2015”.
Ronald Sole’s comment is talking about the unrelated “possessive” or “genitive” construction, which coincidentally also involves the letter S. “2015’s ___” roughly equivalent to “the __ of 2015” (e.g. “this was 2015’s most memorable movie”). It isn’t that common to use the possessive/genitive -’s construction with a year, but it is possible.
The plural “the 1960s” construction can be written with or without an apostrophe. Many current style guides recommend not using an apostrophe, but the apostrophe convention with plural years written as numerals (like “90’s” or “1990’s”) has a significant history of usage and so should not be treated as a “mistake” along the lines of writing “blueberry’s” instead of “blueberries” for the plural of “blueberry”. Unlike with blueberry, there is not a single universally accepted standard for writing the plurals of years. See Sven Yargs' answer to the ELU question Is an apostrophe with a decade (e.g. 1920’s) generally considered “incorrect”?
Absolutely not. The phrase "the nineties" refers to a number of years without specifying exactly which ones. It clearly refers, however, to more than one of up to ten years and so is a plural concept, which justifies the terminal "s." The year 2015 is not a plural concept. The thought you mean to convey can be expressed as
approximately during 2015
during 2015 or thereabouts
There are probably others that do not spring to my befuddled mind right away.
Furthermore, as R. Sole pointed out, if you want to use numerals to express a vague collection of years, the standard way to do so is to use an apostrophe as in
So your usage is wrong as is your punctuation.