I actually wrote my undergraduate thesis partially on this topic. They do not appear to genetically related, as in the Catholic Rosary, the Orthodox/Easter Catholic prayer rope, the Islamic Tasbih/Misbaha, and Hindu/Buddhist/Sikh (japa)mala do not appear to have arise primarily through borrowings. There is certainly some influence between the Hindu and Buddhist traditions, but there's no evidence that they connect to the Western (including Muslim) traditions. There's not even good evidence that the Western traditions are at all related to each other, though it seems that some of them have experienced some morphological convergence since innovation (for instance, the Greek worry beads look strikingly like the Tasbih their Muslim neighbors used). The Eastern and Western Christian traditions of prayer rope and rosary/paternoster may further be connected, but even that is not entirely clear last time I looked into it, and I felt the better argument could be made that they arose independently as objects due to a similar functional necessarily: counting out prayers.

These counting methods seem to arise spontaneously when a tradition recommends a repeated supererogatory prayer (supererogatory acts are acts that are meritorious but not obligatory; it literally means beyond the required). The paternoster (a precursor to the rosary) was innovated when illiterate monks couldn't recite Psalms and instead replaced them with Our Fatherss (in Latin, Pater Noster). Theologians across traditions tend to love the high ideas more than material objects, so the history of these things are poorly attested. Of these traditions, the best attested is the rosary, and the rosary as a cycle of prayers predates the rosary as an object on which those prayers are counted (indeed, the Catholic Church is generally scrupulous about referring to the rosary and rosary beads separately, the later also sometimes being called a "chaplet"). The best evidence suggests the same occurred in other traditions (there's only one good history of the tasbih that I'm aware of, and it's in German--I struggled through the relevant bits when I was an undergrad). The reason why Jews don't have a similar prayer counting system is that there's simply no prayers to count--one doesn't say 18 sh'mas the way one says 33 Hail Marys or Our Fathers. This also explains why there's no Protestant equivalent (though someone did try to develop an Anglican tradition in the last few years).

It seems these develop in literate traditions, ones with set liturgies, but ones where the majority of believers may be illiterate or minimally literate, and thus where repeated supererogatory prayers are recommended, the prayers are simply or memorized. Since these prayers may be hypothetically repeated infinitely, generally recommended cycles develop: 27, 33, 54, 99, 108, 150, whatever (the Dominican rosary cycle is actually rather complex, as it involves unequal numbers of several separate prayers in a set cycle; there are several other rosary-based prayers that don't all use the same material object). To keep track of ones progress in the cycle, a loop of beads or knotted rope becomes useful. Though in some places you see other traditions, such as moving beads or stones from one bowl to another, knots/bears have clear advantages in terms of mobility while praying (you're not tied to the cups) and the ease of traveling with these prayer accessories (you're not afraid of the counting stones spilling).

The similarities appear to be the same solution to the same problem (tracking supererogatory prayer), rather than one tradition emulating a hated rival's. That is, it's a functional similarity, not a genetic relationship.

and the Buddhist situation is even more surprising to me

Can't answer your question about why so many cultures have it, but as to why Buddhism does, that's easy. They inherited it from Hinduism along with a bunch of other symbols and ideas.

The string of beads used to count prayers has a very old provenance in Hinduism, in fact, it goes back to the Rig Veda (1800 BC - 1200 BC) which describes it as a japa mala. The word "japa" means to chant something repetitively, usually mentally or in a low voice if it's audible. The beads were used to count the number of repetitions which are ritually prescribed.

The japa mala commonly used in India has 108 prayer beads plus an index bead (the zero point) and 3 divider beads. Together, the index and dividers break up the 108 prayers beads into four groups of 27. This form of the mala goes back at least to the Aitereya Brahmana, which is the Vedic commentary on the shakala shakha, one of the five branches of the Rig Veda.

Buddhist prayer beads derive from the Hindu mala. The number of beads vary depending on the kind of Buddhism. Tibetan Buddhism retains the full 108 beads of the Hindu mala. Other forms use some even divisor of 108, usually half (54) or one quarter (27). Chinese and Japanese Buddhism generally use the shorter 27 bead versions. Their repetitions include physical activity, not just chanting (like prostrating yourself on the ground with each chant), which is more tiring.

As for why so many cultures use prayer beads, a reasonable explanation seems to be for counting. Humans have been stringing beads for 100,000 years - they are among the oldest artifacts of our species aside from tools. We see evidence of beads used for counting or math in many contexts, for example the abacus. A string of beads makes a cheap, portable counting device and if you close it in a loop as a necklace, you can wear it around your neck and free your hands.

The point of using beads rather than actually counting is because you don't want to get distracted from the prayer or the chant, your focus is on some spiritual activity. Flicking beads with a finger is a mechanical action that doesn't distract, and every time you circle back to the index bead you know you've completed the prescribed number of repetitions.