To me both seem to be just a practical solution to keep track of how far you have gotten in saying/chanting your prayers/mantras
Well, if we are qualifying the nature of our answers, here what I would characterize as a practical sense answer:
It seems that these three major faiths all use prayer beads as an aid for repetitive prayer in which the goal is at least mediation. It is easy to see that when performing repetitive prayer, it helps to have something to keep track of the number of prayers prayed. Hence it makes sense that someone would use something like beads in repetitive prayer because they are small, inexpensive, and easy to make. And repetitive prayer by nature seems to aid meditation, and meditation is important to any religion....
And here is the web site I took that from. https://www.catholiccompany.com/getfed/prayer-beads-phenomenon/
And just an observation, I don't understand why there is such a problem when pagan influences are found in the history of the Church. When Christ died and rose again, he established a new reality that worked backward in time as well, and re-established the relationship of all good with His Father, the Creator. He then sent the Holy Spirit to sanctify what was still in the future.
You don't have to see it my way, but my way is a lot more fun for me that trying to live as you who don't think that makes sense seem to enjoy.