For the same reason that Ona is nominative in Polish as the subject of the understood verb ran, "she" would be preferred in English of my youth but used with much less frequency in American English. As an English teacher, I would have marked "her" as the incorrect case of the pronoun in this sentence.
"than she" isn't from centuries or decades ago. Like the explanation given above by Jellei for Polish, the word "does" is implied but omitted in English also. "I run faster than she (does)". You would never say "I run faster than her does", and so for that reason "I run faster than her" is completely wrong -- if "than" is considered a conjunction introducing a new clause. However, many consider "than" to be a preposition, meaning in a sense "as compared to", in which case "her" is proper English. "I run faster as compared to (than) her". So, both are considered correct. The "than her" version is much more common in current English speech and writing; however, I wouldn't risk writing it that way in a formal document. Most will find it sounds uneducated and wrong.
As a senior editor for the past 20 years in newspapers and online in the UK, New Zealand and Australia I can tell you that if one of our writers wrote 'I can run faster than she' in a sentence I would change this to 'I can run faster than her', as would most editors and explain their reasoning why. The reason for this change is that it's not incorrect to say I can run faster than she, it's just non-standard, non-conventional English that draws attention to itself with how unusual and quirky it is and requires more energy on the part of the reader/listener. The aim, with writing/saying anything for the media or for any other platform, even online, is to write or speak in a way that's the easiest and simplest for the audience to understand. This is called writing/speaking in plain English and is the standard for most situations.
So, in reality, the implied verb in English isn't really "does", it's "runs", similar to the explanation of the Polish "niż ona (biega)" in a later comment.
The reason for that is that the "does" is actually a shortcut for "whatever the verb was in the phrase that preceded the "than". Because, think about it -- "than she does"... what exactly is she doing? In this example, it is the running that I do.
The only reason I'm writing all this is because I tried "... than she runs" and it wasn't accepted.
@gkouye: Okay, technically the meaning is "than she runs" and not literally "than she does", but as you say, it's a shortcut and that's how "does" is used.
I don't recall ever hearing a sentence constructed similarly to "I run faster than she runs", repeating the actual main verb.
I suppose it would have been better to have said that using the objective "her" instead of the nominative "she" would be non-standard English. I'm sure that I could get more than 50% of American teachers of English to agree with me. While a great number of English speakers would agree that "than her" is acceptable, you really would be on shakey ground to say that "than she" is wrong. What is important here is to understand that English pronouns have declinable cases, just as Polish does. Wouldn't it make more sense to use the nominative case of the feminine personal pronoun in English to translate the nominative feminine pronoun in Polish? It really isn't a matter of trying to hold on to decades or centuries old rules. I (not 'me') was just trying to clarify case agreement and making a joke about being an old English teacher and student of the English language who (not 'whom')still thinks case still matters.
It's the exact same in English. "She" is correct, and for most grammarians, it is the preferred pronoun. Some of us get grumpy when "than" is used as a preposition. Personally I think it's sloppy.
This writer is in the other camp, but see for example https://www.merriam-webster.com/words-at-play/than-what-follows-it-and-why
As one of the original commenters on the this topic I find it interesting that it keeps coming up. No one seems to discuss the reasons for why there is confusion. Why does the case of a noun or pronoun change? English has few examples of case changing. Pronouns are the most abused or confused. For nouns the most recognizable is "'s" for possession. He has a dog. It is his. Jan has a dog. It is Jan's. Another reason case changes is following a preposition. He received a book. I gave it to him. They received a book. I gave it to them. He and she received books. I gave them to him and her. "And" is a conjunction. "To" is a preposition. In the case of this topic what is the word "than". Most of the time "than" is a conjunction. Words joined by a conjunction are in the same case. If these two words are the subject of a sentence they are both in the nominative (he and she). Should both of those words follow a preposition, words on both sides of the conjunction should be in the same case (to him and her). "Than" can also be a preposition. So what is "than" in this sentence to be translated. In Polish it seems to be a conjunction which conjoins Ona with the understood "I" of Biegam (I run). If "niż" is a conjunction then there is case agreement. Both are nominative. So what should the translation be? A strict translation should be "she" as that is the nominative form as I stated above. Shouldn't the point of all this be to get a better understanding of the languages and not argue about what sounds better because three years later we are still at it.