I wonder if business cycle fluctuations are usually studied in a (New Keynesian) DSGE model, or can they also be studied in a growth model?
Business cycle fluctuations can be studied in many ways including also growth models. An example of using growth model to study business cycle would be this 1995 paper by Cho & Cooley. This being said growth models are not as common for examining business cycles as for example DSGE models so saying they are usually used for that purpose is correct. To my best knowledge no statistics on the relative use of this or that model for explaining business cycles exist but anecdotally, speaking from attending macro conferences, DSGE or various non-stochastic (when we talk about pure theory) New Keynesian or to lesser extent real business cycle models remain popular.