Levels of formality exist in most (if not all) natural languages, but the difference between what you've described above and a proper grammatical system is that the grammatical system is standardized, accurate and absolute. There are clear rules and clear formality levels involved.

Think about tense for instance. In English you have 3 basic grammatical tenses that can be combined with an aspect (perfect aspect, continuous aspect or both).

Mandarin Chinese has various aspect markers (like 了 le, 过 guo, 着 zhe or 在 zai), but does not have a grammatical marker for tense. That doesn't mean you cannot say that something will happen in the past or future: you just indicate the time with an adverb such as "tommorow", "yesterday" or "in the future".

The reason linguists say that Mandarin has no grammatical tense while English has grammatical tense is that the English tense is highly systematic. There are only 3 basic tenses (Past, Present and Future) and they are always marked the same way. On the other hand, Mandarin is flexible. You can mark the tense by using any temporal adverb you want, or even leave it out completely and let the speaker learn about it from the context. In English you don't get that level of flexibility: you HAVE to mark the tense, and you have only three ways of marking it (12 when you combine these ways with all the possible aspect combinations).

The same idea goes for thing like formality levels, honorifics or evidentiality.

English does have language registers, but there are many ways to express formality, and it's rarely clearly which of these ways is more formal. For instance, you can say:

"I would like to express my deepest gratitude for your invaluable assistance in this matter."

"I am sincerely grateful for the valuable time and effort you have so kindly dedicated to helping me."

"I am eternally in your debt for your gracious help in resolving the matter at hand."

Which one of them is more formal? I don't really know, and wouldn't want to spend a single moment debating about that.

However, in Korean it's pretty obvious that 합니다 (hamnida) is quite formal, but 하옵나이다 (haomnaida) is something so formal you'd usually only hear in historical dramas.

Japanese has less distinctions, but everybody can tell the difference between polite "desu", the polite formal "de gozaimasu", the formal (but not polite) "de aru", the informal and familiar "da" and then dated polite "de gozaru" (which you'd mostly hear in Samurai movies or other period dramas).

The situation in which you'd use each levels are also well-defined than in English. In particular, the familiar version is not neutral. You don't just use the Japanese "da" with people you don't know very well. It's not slang at all, but it's still quite rude to use it with strangers or your superiors.

And English has nothing quite like the distinction between "de aru", "de gozaimasu" and "desu". In most cases where you want to be polite (not formal) you would use "desu". "de gozaimasu" is used in certain settings that require extra formality (e.g. when talking to customers or in public announcements). "de aru" is something you would usually not use when addressing to people directly, since it is too plain (i.e. it would be quite rude). Instead, you find it in encyclopedias, reference books and documentaries.