Is it illegal or unethical to use plotting and graphics styles that mimic those from newspapers and websites? I am a researcher and publish my findings in academic journals, websites/blogs, and social media. The results often involve plots and other graphics, and I try to make them as attractive as possible.

Publications such as NYT, FT, The Economist, ... have characteristic (and pretty imo) styles for plots that many people like and try to mimic. One particular example is fivethirtyeight, with their style being even integrated as a style in matplotlib.

I wonder what is the stance about this topic in the data science community and academia. Is it okay to get inspired by someone's design of graphics and follow it?

I am not sure whether a plotting style can be protected and patented (and can you get sued for re-using it), but if not: is it unethical to mimic plot styles from others? 

Another thing that may complicate things further is that when one publishes in academic journals, often the copyright of the content is transferred to the publisher.
It's fine to base your graphical style on that of other publications that you think are doing a good job.  There are no ethical issues with this, provided you are not actually reusing images—or parts of images—from other sources.  (For example, you cannot copy parts directly from other figures without either proper attribution or permission; what precisely you need depends on the context and the use you are going to make of the images.  You might also run into issues if you directly copied The Economist's proprietary house font, which was specifically designed for that publication.)