I'm not sure there is any such thing as a science of language. If anyone knows, I'd appreciate some references.
The scientific investigation of language is called Linguistics. I have a master's in the subject, with a specialization in Sociolinguistics, which is the study of language as it is used in its various social contexts, so the question you have asked here is probably closest to the sort of question I would look at.
This summer I read the book "Ministers of Reform" by Robert Crunden, which has several historical profiles of American Progressives. One of his claims is that American Progressives adopted religious language for use in the Progressive cause even though it was largely a secular movement. He further intimates that religious language has long been a part of American political discourse. His argument was persuasive, but at the same time the definition of "religious language" is rather vague. Proving such a point depends largely on testimony of the people involved that they were aware they were using religious imagery.
Does it? I don't necessarily think people need to be conscious of the fact that they are using language rooted in religion to be able to make the case that they are doing so. For instance, in most varieties of Arabic the word for "Saturday" is Sebt سبت, an obvious cognate (word coming from the same root) with the Hebrew Shabbat (both of which mean the same thing -- 'rest'), from which we get the English "Sabbath", for the Jewish day of rest specifically. While I'm sure if you asked any native Arabic speaker about the etymology of their word for "Saturday", they'd be able to tell you that it means 'rest', they would not necessarily connect it to the specifically Jewish concept of Saturday being the "day of rest" as described in the Hebrew Bible (Christians probably would, since they know the Genesis story, and of course Jews definitely would), because of course most Arabic speakers for the past 1,400 years have been Muslims, and in Islam Saturday is not the day of rest/the day in which no work takes places and you perform instead your religious duties --
Friday is. (And this
religious fact is reflected in the Arabic for "Friday", which is usually youm el gum'aa يوم الجمعة 'day of gathering', referring to the fact that Friday is the day of the week when Muslims in particular gather togther for their corporate prayer in mosques; cf. other days of the week, which are numeric: youm el ahad 'first day'/Sunday, youm al ithnayn 'second day'/Monday, youm el thalatha 'third day'/Tuesday, etc.)
But are people always aware of the roots of the imagery they use when speaking?
No. See above.
If not, would it still be "religious language"?
It could be argued to be, sure. I mean, I just argued for an example where I believe it is, and I think it is fairly easy to see how it is in that case, since it has to do with the actual etymology of the words used. Over time, etymologies may be forgotten, other (folk) etymologies may be embraced in their place, etc., but some are plainly obvious by cross-linguistic comparison such that even the majority of the speakers forgetting about that connection doesn't really do anything to the truth of that connection, as in the Hebrew/Arabic case given above. (Generally speaking, the more 'basic' a term is seen to be cross-linguistically -- and days of the week are pretty basic, seeing as how almost every culture has some conception of days being given some unique names in relation to each other, even if they don't have a standard western/European 7-day week model -- the stronger these connections are, because the more basic the term to the organization of the world and things in it, the more often it is used and hence its meaning/s reinforced. This is why certain word categories are much more stable over time than other word categories: <traditionally> pronouns, numbers, adjectives of size, etc. They aren't like nouns where we're adding new ones by the bucketful all the time:
internet, search engine, 'expanded universe' in the film sense, etc.)
Can other types of language be identified: scientific language, political language, etc.
Certainly. Plenty of people have built their careers on studying these things. Linguists like George and Robin Lakoff come to mind.
Is a science of language possible?
Not only is it possible, it's arguably one of the oldest, most well-established sciences that there is. Whereas the scientific method itself in its most commonly-known form is often tied to the rise of rationalistic thinking via the European Enlightenment period of the 17th and 18th centuries AD, the earliest grammars of language that we know of came out of India in the 6th century BC, with the work of Pānini. The Greeks and Mesopotamians also formally described their respective languages and systematized the study of them in the pre-modern era, while the linguistic study of Arabic, Hebrew, Syriac, etc. developed later in the Middle Ages (see, e.g., the famous Persian grammarian Sibawayh for Arabic, or Jacob of Edessa for the equivalent groundbreaking work in Syriac). All of these were centuries before the development of what we think of as the modern scientific worldview.
Granted, 'modern' linguistics is rather new (as the name would suggest), and sort of grew up in tandem with modern Biology (we use a lot of same the terminology, e.g., "family", "genetics", etc.) beginning in the late 18th century. Just to put a framework on it, one of the classic linguistic texts I was given to study as an undergraduate was Ferdinand de Saussure's
Course in General Linguistics, which was published posthumously by his students in 1916 based on notes from lectures that their professor had given in Geneva, Switzerland.
I don't know for sure, but I get the impression that the 'modern' versions of basically any science usually don't usually begin until the late 18th century at the earliest (maybe technologically we were not able to do a great many things until that point; I don't know), and generally involve the gathering together of what had been up until that point separate disciplines/pursuits (in linguistics' case, its most obvious direct precursor was probably philology, which is the study of language in historical settings from collected writings, speeches, etc.; hence the earliest modern linguistic texts tended to involve historical comparison, what we would today consider most properly the domain of the Historical Linguistics subfield; I suspect this is what made de Saussure's work so special for its time -- it concerned
general linguistic principles, rather than the historical study of, say, how the modern Romance languages evolved from Latin, or the remains of Arabic in Spanish and Portuguese, or whatever).
I hope this helps, OP...at least until Dave-W's friend can write and prove how wrong I am with my lowly lesser degree.