Even Muslims do not claim their religion is completely novel. The rituals may differ, but the creed preached by Muhammad (peace and blessings upon him) is the same preached by Adam, Noah, Abraham, Ismail, Isaac, Jacob, Joseph, Moses, Aaron, David, Solomon, John, Jesus, and countless other prophets (peace be upon them all): worship The One True God (Allah, Yahweh) with no partners.
As far as prayer rugs and other accessories, those are not actually part of our rituals as Muslims. Some people use them for practical purposes (prayer rugs help you avoid prostrating on dirt, asphalt, a potentially unclean carpet, etc. and prayer beads make it easier to keep count), while some others may have cultural reasons, and some just want to enhance their spiritual experience (e.g. incense and perfume).
Muslims say that Islam _was_ the religion of all the prophets you've mentioned. Because of this they also believe that Islam is the _oldest_ religion, since Adam, the first man, followed Islam.
In fact the Prophet Muhammad SAW did not use a prayer mat and instead he placed his head on the soil (ground/earth).
Also note; Shias place their head on a stone/rock to mimic this practice: https://en.m.wikipedia.org/wiki/Turbah