First of all, congrats on getting into the second round with McKinsey! That's amazing :)
Secondly, thanks for asking your question in our Consulting Q&A. I found some old threads that might include useful information for you.
The first one is about Bain, but experts explain the differences between first and second rounds in general. Let me quickly give you excerpts of the answers provided by our experts:
Chern (passed both interviews with McK and BCG):
The key differences between the 1st and 2nd interview rounds, besides the interviewers being principals or partners as you mentioned, are in structure, focus and selectivity. The structure in the 2nd round will most likely not follow the standard fit + case interview format. Partners may prefer to launch into a conversation about the latest developments in a particular business, industry or geography and talk about your motivations and life experiences later [...]. The focus in the 2nd round will likely be influenced by your performance in the previous round. The 2nd round interviewers will test on any areas where they perceived you needed to work on [...]. The selectivity in the 2nd round is much lower, as this is the round where hiring decisions are made [...].
Hemant (Ex McKinsey):
There is practically no difference between 1st and 2nd rounds. [...] Some R2 cases have more ambiguity but that's only because a partner/principal may not have time/patience/interest to give a typical case so they make it up as they go. It also makes the interview more fun for them, not having to type out notes and check for your answers vs the one provided by the HR.
You can find the original Q&A here: Bain&Company second round interviews
Another Q&A particularly addresses the difference between first and second rounds in general.
Matteo (referring to McK, Bain and BCG) mentions 2 points:
1) during the first round the interviewers will be senior managers and/or junior partners, wheather in the second round you will generally meet senior partners or directors
2) Therefore since the two profiles mentioned above have a different seniority during the first round it is more likely that you will be asked more analytical questions and the business cases will be more detaild (e.g. estimate the market share at the endo of the third year knowing that.. ecc.), on the other side in the second round the you might have to tackle more broad and strategical questions (e.g. what would you outsource in a marketing department)
You can find the original Q&A here: Difference between first round and second round interviews?
I hope this helps!
Best of luck for the rest of your prep and of course, the second round!