1 / The grammatical category is a system of opposed rows of grammatical forms with homogeneous meanings. The grammatical categories in their complex relationships with each other constitute the core of the grammatical structure of the language. (http://www.langust.ru/rus_gram/rus_gr01.shtml#rg01_04)
2 / In the studied languages, the comparative frequency of 21 characteristics was assessed, among them word order, types of sounds, ways of forming negation and much other. This made it possible to find languages that combine the largest number of relatively rare characteristics - that is, the most unusual languages of the planet. (http://www.langust.ru/news/28_06_13.shtml)
So, let's start with what a "grammatical category" is. A grammatical category is a collection of forms that express a certain grammatical meaning. The category of time, for example (in Russian there are three of them: past, present, future), or the category of number (in Russian there are two: singular and plural). A language cannot "obey" a grammatical category; on the contrary, these are certain grammatical categories that may or may not be present in it (in Hungarian and Japanese, for example, the category of number is absent, in Armenian - the category of gender).
And still: are there phenomena and patterns common to all languages? Yes, there are: for example, all languages have parts of speech (nouns, verbs). Such phenomena are usually called "linguistic universals". Some believe that the existence of linguistic universals proves the origin of all languages in the world from one proto-language, or that some basic grammatical structures are genetically programmed in the human mind. These provisions are developed and explored by the so-called "universal grammar".