If I have two Thorncaster Slivers, does their effect stack? That is, do the other slivers I have get the "1 damage" ability twice?
If not, why? What rule makes it so they can't have the ability twice? If I had a different card with same ability would they get it twice? What if the different card had a slightly different wording? Or caused 2 damage instead of 1?
In what situations would a creature be allowed to have two copies of the same ability (if ever)?
Yes, they will have two copies of the ability:
112.2c An object may have multiple abilities. If the object is represented by a card, then aside from certain defined abilities that may be strung together on a single line (see rule 702, “Keyword Abilities”), each paragraph break in a card’s text marks a separate ability. If the object is not represented by a card, the effect that created it may have given it multiple abilities. An object may also be granted additional abilities by a spell or ability. If an object has multiple instances of the same ability, each instance functions independently. This may or may not produce more effects than a single instance; refer to the specific ability for more information.
Two copies of the triggered ability will mean two damage whenever a sliver attacks.
As others have said, yes, the ability will exist twice on each of your Sliver creatures, and each of those creatures will trigger it twice upon attacking.
In what situations would a creature be allowed to have two copies of the same ability (if ever)?
There is no limit to the number of copies of the same ability a creature can have. They are each added to the list of things that creature can do, and triggered abilities will each trigger separately. If you have 4 Thorncaster Slivers on the field, each sliver's attack will cause it to deal 1 damage 4 times. Notably, this allows you to distribute your damage however you like--you are not limited to dealing damage in packets of 4.
This doesn't mean that multiple copies of the same ability are always useful. For example, it's stated in the rules that multiple copies of flying, haste, lifelink, and many other keyword abilities are redundant, so a creature that gains flying twice doesn't have any advantages over a creature that has it once, and a creature with two instances of lifelink doesn't gain you twice as much health (although cards like Armadillo Cloak, which don't use the keyword but have similar effects, will stack with lifelink). See this list for which abilities stack and which don't.
Also worth noting that in cases where the duplicated abilities have a cost, such as Prodigal Pyromancer being enchanted with Arcane Teachings, it doesn't simply increase the effectiveness of the ability. Prodigal Pyromancer will have 2 copies of the "{t}: deal 1 damage" ability, but each of those costs must be paid separately--tapping once will allow you to activate either the innate ability or the one granted by the enchantment, but not both. Triggered abilities which have a cost (e.g. "When X happens, you may pay Y. If you do, Z.") must also be paid for separately.
Yes, you get two instances of the ability.
You can also reference the _Gatherer page for Thorncaster Sliver which has official rulings.
If you control multiple Thorncaster Slivers and a Sliver you control attacks, each ability will trigger separately.
If you change the creature type of a Sliver you control so it’s no longer a Sliver, it will no longer be affected by its own ability. Its ability will continue to affect other Sliver creatures you control.
Abilities that Slivers grant, as well as power/toughness boosts, are cumulative. However, for some abilities, like flying, having more than one instance of the ability doesn’t provide any additional benefit.
