Congrats on getting past the first round!
First: Don't assume/expect anything in particular. The partner round could be exactly the same as your first round or completely different. That said, be prepared for them being less structured/linear, more "brainstorm-y", more intense, etc. Additionally, make sure to have worked on any key weaknesses that were on display in the first round!
Second: You should be more prepared for a pretty in-depth fit interview. While I'm going to include tips below on why consulting, why the firm, resume walkthrough, and fit stories, I highly recommend you expect "loose" or "random" questions and practice this scenario with a coach. I would venture to guess they won't just ask you 20 "tell me about a time" questions. They may have a bit more of a chat, ask you to brainstorm, ask things like "How would you setup x type of project", etc. etc. You need to prepare to be flexible here!
Preparing for Personal Stories
Step 1 - Categorize the main stories
There are 5-10 "themes" you need to prepare for. i.e. Leadership, teamwork, challenge, etc. Figure out this list and make sure your stories cover this range (PM me and I can provide you with a template for this list)
Step 2 - Create FLEXIBLE stories that cover a range of categories
You need to create 4-6 stories that each cover a range of topics. They need to be powerful stories that can be adjusted and adapted based on the question asked.
One of my "core" or "killer" stories was usable for Initiative, Achievement, Leadership, Challenge, Change of direction, AND Persuasion.
Write down these stories along STAR or similar format...use bullet points
Step 3 - Organize these stories so you know which ones can be used for what and PRACTICE
Make sure you cover the whole gambit. Then, practice getting asked a question and thinking of which stories apply. I can assure you, no-one is coming up with full stories in a few seconds. Rather, they have practiced how to adapt an existing story to the question asked.