The Census Bureau, the data angency charged with creating statistics on household incomes and their distribution, does not maintain an official definition of middle class status. This is because there is more to the definition than just strictly income; social factors also play a role. Thus, for its definition, the following six indicators are suggested to gauge whether you are middle class.
If you specifically refer to the verb "maintain", here it is used in the sense of "express" https://dictionary.cambridge.org/de/worterbuch/englisch/maintain
In your passage, maintain means assert or declare (or—more simply—have):
The Census Bureau . . . does not declare an official definition of middle class status. [because there is no way to do that based on income data alone]
In the context, it’s basically saying that the Census Bureau don’t publish a formal definition of the term “middle class” that can be applied to their statistical data.
Assuming we’re taking about the United States Census Bureau, they publish definitions of all of their technical terms here:
As one of the nation's key statistical agencies, the Census Bureau collects data on many topics vital to people and businesses. The Census Bureau adheres to scientific methods and therefore often uses precise technical terminology that may be unfamiliar to the public. Listed below are some of the major glossaries from our data programs to help people who wish to understand and use our data.
The original quote is claiming that ‘middle class” is not one of those technical terms.
The word “maintain” is probably being used with this definition in mind:
to continue to have; to keep in existence, or not allow to become less