That whole bundle of traditions - prayer times, prayer directions, prayer accessories, etc predates Islam if for no other reason than that Islam is comparatively new. But the cross-pollination could have easily happened more than once and in both directions.

In fact the Muslims initially prayed towards Jerusalem (as did the Jews then and today) until revelation specifically turned them towards Mecca:

> "So turn your face toward al-Masjid al-Haram (the Sacred Mosque in Mecca). And wherever you are, turn your faces toward it..."

Quran 2:144

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).