I have several points which appear duplicates in the feature matrix (same values for the features). These points may have different values of the target variable. What is the appropriate way to handle this case during train test split? Should I just ignore the problem and go on with random split or remove duplicated points ?
It depends on what does a duplicate mean in your data.
If your features contain identification variables (for example, transaction number, student ID, etc), then duplicates are just copies of a same sample. In this case, if their target values ($y$) are the same, you should remove the duplicates. If they have different $y$'s, then this sample is problematic, and you should probably consider removing them altogether.
If your features does not contain identification variables, then it is totally possible that samples with the same feature values are different samples, and therefore possible to take different $y$'s. In this case, you don't need to do anything special.