> Makes me wonder if there's some sort of grammar that's innate to the human brain.

Yes, but what it consists of is a perennial source of controversy. :-)

https://en.wikipedia.org/wiki/Linguistic_universal

On the one hand, there are lots of concepts that are found in lots of different languages (so understanding a concept in one language is a big help for another language). Noun case is a great example of that, but it's use to different extents in different languages, and languages that have noun case don't always use it as extensively as one another (for example, in Latin it's arguably marked directly on every noun and adjective, even on names, while in English we nowadays only have it in pronouns!).

Also, languages have different numbers of cases (some people say that English just has subject and oblique, though perhaps also a genitive, while German still distinguishes a dative, Latin also has an ablative, and Finnish and Estonian famously have dozens of cases).

You definitely can't use noun cases as a linguistic universal, though: for example, Chinese doesn't have them at all. There Barney's "I love you, you love me" should just be
"我爱你，你爱我" (I love you, you love I). Whereas in Latin it would most idiomatically¹ be "ego te amo, tu me amas"—every single word is different between the "I love you" and "you love me" cases, and the grammatical relationship is still also reflected (although not explicitly expressed²) by the word order.

So it's one thing to benefit from learning grammatical concepts that you can apply to new languages as you learn them (as well as to your native language), and another thing to expect to encounter each concept in each new language you look at, which is most likely not going to happen.

There are also two different phenomena which are likely to make some grammatical features feel recognizable:

https://en.wikipedia.org/wiki/Language_family

(the fact that many languages are historically related to one another) and

https://en.wikipedia.org/wiki/Areal_feature

(the way that languages that are unrelated to one another by descent nonetheless influence each other through language contact within a geographic area, or sometimes trade or religious influences)

Many of the languages that you encounter in Europe, in former European colonies, and in Central and South Asia (although not Hungarian!) are part of the Indo-European family.

https://en.wikipedia.org/wiki/Indo-European_languages

These languages share a common descent (and include all of the Romance and Germanic languages, and many, many others). They often have features like prepositions, participles, noun inflections (like case markers), verb inflections (perhaps tense, number, mood, voice, and person, although not all languages include all of these distinctions), and agreement (certain forms of words have to match certain forms of other words in the same sentence in order to confirm a grammatical relationship).

The Indo-European influences as well as areal effects have also resulted in a phenomenon sometimes called "Standard Average European"

https://en.wikipedia.org/wiki/Standard_Average_European

where European languages are, by worldwide standards, grammatically rather similar to one another. But these similarities shouldn't all be taken to be linguistic universals that reflect how the human mind works, because many other languages don't have them, and may also have other features that are rare in European languages. (Some examples that are somewhat common in languages overall but rare in European languages and in the Indo-European family include ergativity and evidential markers, while marking number on nouns or tense on verbs is less common in languages from some other parts of the world.)

¹ This example uses subject pronouns, even though they are optional in Latin (unlike English). https://en.wikipedia.org/wiki/Pro-drop_language

² This example uses SOV order, which is the most common order in Latin but isn't grammatically required at all due to the noun case markings. For example, "te amo ego" is less idiomatic but is still grammatically acceptable and still has the same meaning.