What do you mean by verb "respect" in Slavic languages? It is an English word.
My respektujeme tamtu ženu. - accusative
My si vážíme tamté ženy. - genitive
My si ceníme tamté ženy. - genitive
My oceňujeme tamtu ženu - accusative
You cannot think about English verbs here, it is not useful. You must think about Czech verbs.
I'm thinking about the equivalents of "respect" in all the languages I know. For instance in Polish you can use a couple of verbs for this or similar concept (szanować, respektować, cenić sobie) but all of them take the accusative case, unless one goes for a descriptive way ("to have respect for" + genitive indeed).
Besides, everyone intuitively thinks about the verbs in their own language when learning a new one, it's a natural process.
You learned many more languages than me but in those that you achieve some decent level I believe you will agree that you cannot constantly translate stuff to another language. You must understand the expressions and sentences directly. The more the language is different, the more it is important. I think that even too much direct comparison with our original language is unhelpful.
Currently, when I am learning Latin (not just here), the way verbs are used is really different, their meanings are often very broad and they can take strange cases. Even at a very rudimentary level it is better to try work with Latin expressions only. Modern Latin textbooks are single language and it is often recommended not to translate the sentence before understanding it.
All the examples I showed were some equivalents of "respect". And I expect some o them to exist and take genitive in other Slavic languages as well. But I can be wrong in this regard. Other take the accusative in Czech, most notably "respektovat".
Certainly there are other Slavic verbs that take other cases like Russian дорожи́ть (instrumental).
You can't be thinking in a new language where you're on lesson number 10 or 15 or so. Come on! It's obvious that people compare languages, especially the ones from the same family so when someone knows English and Spanish and is starting his adventure with Italian, he'll naturally be using primarily the Spanish structures adjusting them by the changes intruduced by the newly learnt material. If these fail, he'll seek comparisons in English. It's not only my way of learning, I'm sure everyone does it subconsciously. That's how the human brain works. It needs points of reference. And it explains why I wasn't the only one to fall in the trap of confusing singular of a different case with plural of the intuitively expected case. Making mistakes and digging for reasons behind them is probably the best way to learn languages.
edited repsonse in light of feedback: the hover hints gives probably the actual czech word used in this sentence which leads learners to believe something is fuzzy with how to state the sentence in english. since i am only a beginner-beginner, i can imagine the concept of 'respect' as a manner of 'weighing' (a person) like putting their character in the balance and finding the character heavy on the 'good' side. this is all complete speculation on my part, but would make sense in light of my ignorance.
By the way, the New verbs note in the Family 1 Tips alerts us to vážit si as requiring genitive.
Czech essential grammar (Naughton) "Certain verbs, especially reflexive verbs are followed by the genitive case, where an English speaker might expect the accusative, " p197