How should I choose the name of my past degree in application when the options do not not exactly match my degree?
By "I see things like," I assume you mean "the options are," i.e., the list of options you have given is exhaustive. I shall also assume that you are an engineering student, with a degree at "Bachelor of science" level and you are currently studying at "Master's of science" level. The question about academic background is looking for your highest qualification, so I would list "master's of science."

You could call the university to confirm!

EDIT. I do not agree with compass's remark that a UK MEng degree is provided in lieu of a Bachelor's and has the same study length: UK BEng degrees are typically three years, whereas UK MEng degrees are typically four years. It is possible that compass is attempting to compare Bachelor's degrees between different countries.