"We have many of this" is NOT accepted and I don't think it is correct English.
"Tyhle máme mnoho" is incorrect Czech. It does not mean anything.
"We have many of these" would be "Těchto/těchle máme mnoho.".
"We have many of those" would be "Tech/tamtěch/tamhletěch máme mnoho.".
"We have many of them" would be "Máme jich mnoho.".
How many people will be grateful to learn from you in future, just as me, Vladimir! Keep being scrupulous, it's the only route to go to reach a satisfying level at Czech. More , don't take my certainties for "real certainties". It happens not to know how to put the question. A teacher once said A good question is half of the answer. If my questions are confusional or misplaced formulations, it means that the confusion is more than one only issue. Unfortunately, sometimes on English and Czech questions at the same time! Having that said, does your first answer (the NOT answer) mean that "a lot of" and "many" are used in different contexts? They aren't completely synonymous, right? Forgive my need for patience and obviously thank you very much.