Are there any examples of languages in totally different parts of Earth having similarities in such things as grammar and words? As one example, I've noticed 'Mama' and 'Papa' seem to be common across the globe, though that might just be due to colonisation.
Some words are very consistent across the world for non-linguistic reasons. For example, "mama" consists of the earliest sounds infants are able to make. This is the same reason languages on opposite sides of the world have words that sound like "mow" or "meow" for "cat".
Some linguists have also proposed that particular sounds or groups of sounds are fundamentally associated with particular concepts in the human brain, though this hasn't met with wide acceptance.
Words can also be borrowed from one language into another; this is especially common with words for trade goods or new technologies, which can easily become so-called "Wanderworter" (or Wanderwörter if you're German) spreading across the world along with the things they describe. That's why so many languages have similar words for "tea", and also for "internet". Cognates of "cannabis" appear all the way back in ancient Sumerian.
Grammatically, certain structures are extremely common across the world, which some linguists attribute to "universal grammar" built into the human brain. For example, languages that put verbs before their objects, also tend to put nouns before relative clauses. When one of these features changes in a language, the other tends to change with it.
Grammatical structures can also be borrowed between languages. A lot of languages in western Europe have the same structure for expressing "perfect" verbs (ones that describe the aftereffects of a past action): English says someone "has eaten", while in French, someone a mangé ("has eaten"), and in Greek, ekhei phaei ("has eaten"). This is an "areal feature" that has spread between languages due to a long period of contact; when languages are in long enough contact to accumulate a lot of these shared features, they become a "Sprachbund".
And of course, some things end up converging due to sheer coincidence. There are a lot of languages in the world, and only so many different ways to arrange words. English and Akkadian and Swahili all put prepositions before nouns, not because of some ancestral relationship (or any direct contact), but because there aren't that many different places to put prepositions, and "before the noun" is a common choice.