Based on this definition, why economics relating to social science, and what is other than social science?
There is actually no clear cut consensus on where Economics belongs (although it is fair to say most would likely put it into category of social science). Some authors consider it to be science, some social science, some even moral science, and some even argue it should have its own category. A good paper that tries to answer this question for economics is Hudson (2017). As Hudson writes:
Is economics a science or a social science? Arguments have been made for both orientations, i.e. that economics is a social science (Frey 1999), a science (Frey 1999) and even a moral science (Schabas 2009). In terms of sciences it has been linked with physics, with many physicists doubling as, or transforming into, economists and a subpart of physics, econophysics, specifically evolving which deals with economics (Stanley et al. 1999). There are also links with biology (Marshall 1920; Daly 1968). Frey (1999) argues economics is a social science as it is part of those sciences which deal with actual problems of society, i.e. it is the subject matter which makes it a social science. However, he also points out that in practice economists tend to fill the journals with axioms, lemmas and proofs, i.e. they adopt what they perceive as a scientific, and particularly a mathematical, methodology. Mayer (1980) argues that economists act as though economics is a hard science and this is reflected by their sophisticated use of mathematics. However, he then goes on to argue that econometrics is not sufficiently advanced to enable us to test theories as a hard scientist would.
But this being said, as mentioned in my first paragraph most would call it social science as its main focus are societal issues (although you can have 1 person Robinson Crusoe economy, typically economies involve large groups of people).
Why he can make the claim above regarding Type I and Type II in social science?
I do not think there is any rigorous defense of such claim. In any science if you care about the science itself both type I and type II errors are equally problematic. You should ask your professor for why he thinks that, but I do not think that you can rigorously defend such position.
I guess in some specific circumstances you could defend the claim. If we test if some food is poisonous then you should do everything you can about avoiding type II error (e.g. saying the food is not poisonous when it actually is), since if you commit type II error you get someone poisoned, whereas if you commit type I error you just think some safe food is poisonous, but assuming you have plenty of other food options that you know are safe you wont starve and you won't kill anyone.
You could probably set up similar policy scenarios in the subfield of policy economics, but extending that claim about type I and type II error to all social science or even all economics is hardly defensible (if anything I can think of far more examples from medicine which is not considered social science).