By the time of the Babylonian exile (circa 590 BCE), Yahweh was definitely being worshipped as a monotheistic deity. For example, even if one subscribes to a division of the book of Isaiah into two parts, it is widely agreed that proto-Isaiah (chapters 1-39) was written before the Babylonian exile and it is strongly monotheistic.

Yahweh looks to have been solely associated with the monotheistic religion of the Israelites. Some have posed otherwise based on similar names in the Ancient Near East (e.g. the Akkadian personal name Yaum-ilu), but by and large these are not convincing to scholars. One theory, based on an extrapolation from the book of Exodus, suggests that a Yahweh cult was originally established by a clan of the Midianites which Moses then joined (based on the fact that Jethro, Moses' father-in-law seems to officiate at a sacrifice to Yahweh in Exodus 18:10-12). However, this is shaky ground to base any solid assumption on, and at best only explains where the name came from, not that the cultic practices were adopted by the Israelites.

Other names of the god of the Israelites lend more support to the idea that their deity was originally different. The name "El" (אל, often translated God) has, by some, been seen to refer to a Canaanite deity. Texts at Ras Shamra support the idea of the chief deity of the Canaanites being named "אל" and being the father of the gods. However, the waters are muddied when one considers that אל's core meaning is something akin to "power" or "strength", and there are examples in scripture of phrases like "the אל of my hand" which is universally understood as saying "the power of my hand", not "the chief Canaanite deity of my hand."

Of more interest is the epithet "El Shaddai" probably meaning "god of the mountains." Some argue that this descriptor links the god of the Israelites to the Canaanite storm god Baal. This is tenuous, however, because a god being linked to the mountains is hardly unique. Think of the number of religions which place their gods upon mountains and it's easy to understand why this isn't a strong argument, especially when one considers that Yahweh's prime theophany to Israel is upon Mt. Sinai.

It is not unreasonable to suppose that the Israelites adopted some terms from the Canaanite religion such as labelling their god El Shaddai, but to move from that to suggest that Israelite religion started from a local storm god is a large stretch.

tl;dr Yahweh is specific to the Israelite religion, monotheism was well and truly established before the Babylonian captivity, and other arguments that the Israelite religion grew from a Canaanite deity are not well founded and are reaching.

Sources: The Interpreter's Dictionary of the Bible: God, names of (B.W. Anderson.)

The Prophecy of Isaiah (J. Alec Motyer)

edit: The primary lexicon for biblical Hebrew (Brown-Driver-Briggs) suggests that the primary meaning of Shaddai is might/power, and it's meaning of mountain is archaic and secondary, meaning that it's more likely that the Israelites were simply using the word without any connotation to the Canaanite god.

I am not an expert, I have my undergrad in religious studies with a Judaic concentration. I'm currently reading the following book by Smith. This book answers part of your question in where Yahweh's and the God of the Bible's origins come from but it seems a little more complicated. It's a great book that I'd recommend and he cites pretty much everything. I'm not sure of general consensus, but based upon Smith's book I would say it's a normative view among scholarship.

I am at a loss how to articulate some arguments for and against as it seems exceesingly complicated and I'd most likely be copying paragraphs for you to read. I'm also on my phone and about to go to bed. I can try tomorrow to be more articulate. Mods feel free to delete my comment.

The Origins of Biblical Monotheism: Israel's Polytheistic Background and the Ugaritic Texts by Mark S. Smith.

Mark Smith writes in Early History of God that the Old Testament shows signs of cultic or unofficial worship whereas El was worshiped officially but after the Babylonian exile, when the Torah and Talmud was written, the texts were redacted and edited to make it look as if Yahweh and El and El Shaddai were the same god (whereas they were members of the same pantheon which explains some of the bizarre errors and inconsistencies in 1 and 2 Kings).

I read somewhere (and it's been bugging me) that Yahweh was an Edomite war god of a soldier cult that migrated with soldiers to Judah where he merged with storm and sea monster-god Yam/Yaw/Yahu.

William Harwood's Mythology's Last Gods asserts that Yahweh was originally a volcano goddess called Yahuwa based in northern Assyria (current eastern Turkey and South of Mount Ararat) which moved into Canaan. I did find Yahuwa interesting but not compelling.

A good community to answer this question would be /r/AcademicBiblical