I have always found it odd that some people insist on the idea that the Western world is built on Judeo-Christian values. My understanding of history is limited, but it has always seemed to me that the modern ideas of humanism and human rights came about as mostly a response to oppressions and wars caused by the church. I find it even more odd that some conservatives today try to put emphasis on the fact that Christianity shares values with Judaism, when it actually seems that Judaism has more in common with Islam than with Christianity. That makes me wonder - is this idea of Western Judeo-Christian values mainly just an attempt to create distance toward the newest of the big Abrahamic faiths?
This is an intriguing theory - and you're on the right track that 'Judeo-Christian' is a recent concept, and does establish a sense of community and continuity. But it does not gain prominence through an effort to distance Judaism and Christianity from Islam; rather, it separates Western identity from atheistic communism.
Nietzsche does use the phrase, reflecting his perceived sense of continuity (but this is derisive, as this relates to ressentiment). More recent scholars point to earlier usage by F.C. Baur, who does use the term to classify particular Christian sects; and that, in the nineteenth century, there was a relationship seen by both Jews and Christians between Judaism and Islam. Arthur Cohen wrote a provocative essay, "The Myth of the Judeo-Christian Tradition," in 1957, placing the proliferation of the term in a specifically American context against fascism and more saliently communism. It's worth considering that the motto of the United States - In God We Trust - was not adopted until 1956.
However, Cohen was writing both nearly seven decades ago and in a particular American, Cold War context - certainly not how Baur and Nietzsche use the term, and one that may not necessarily reflect more modern anxieties like the one you propose to other Islam rather than atheism, nor its use for the finer theological question of early Christianity.
Judaism has more in common with Islam than with Christianity.
I am not a historian of religion so I am afraid I can't comment thoughtfully on the particularities of any of these religions, except to extend the same caution you propose for 'Judeo-Christian' concepts to 'Abrahamic religions' more broadly. There are also several difficulties surrounding the hyphen itself - does it actually connote unity, or does is separate and distinguish them?
Emmanuel Nathan and Anya Topolski's edited volume Is there a Judeo-Christian Tradition? has several papers addressing the various stakes of this question from the European (rather than American) perspective.