This is a nice example of a problem where half the work goes into figuring out useful structural properties of the solution, and then the other half goes into writing code to search for a solution based on this knowledge. In this case, let $M$ be the maximum number of characteristics any two cows have in common, over all pairs of cows. We claim that $M+1$ is the answer.

Here is a simple argument. If we pick two cows (say $A$ and $B$) that have $M$ traits in common, then we can ask about just those traits, generating $M$ "yes" answers and leaving a feasible set that still contains $A$ and $B$. Hence, the maximum number of "yes" answers in a transcript could be larger than $M$. On the other hand, the number of "yes" answers in any transcript cannot be larger than $M+1$. Suppose we have a transcript involving $M+1$ yes answers after which there are still multiple cows in our feasible set. If so, those cows must have had more than $M$ traits in common, which cannot be the case! After $M+1$ "yes" answers, we therefore must have reduced the feasible set down to at most a single cow.

Now that we know all we need to do is compute $M$, the coding part isn't too bad. My code below does this in perhaps the most straightforward way, looping over all pairs of cows and for each, checking the number of characteristics in common again with two "for" loops. The input size is small enough that this runs fast enough to pass all test cases.

As a note to those en route to silver, however, one could make the "num_common" function quite a bit faster using fancier data structures like a binary search tree (e.g., a "set" in C++) or a hash table (e.g., an "unordered_set" in C++). To compare cows $A$ and $B$, we would add all of $A$'s characteristics to the data structure, then do lookups for all of $B$'s characteristics, counting the number that succeed. Alternatively, we could have sorted every cow's list of characteristics, and to compare $A$ with $B$ we enact the process of "merging" their sorted characteristics into one larger sorted list, a standard algorithmic procedure (e.g., part of the "merge sort" algorithm) that allows us to easily count duplicates along the way. Alternatively still, after sorting, we could run through all of $B$'s characteristics and binary search for them in $A$'s sorted list of characteristics.

#include <iostream>
#include <fstream>
#include <vector>
using namespace std;
 
int N;
vector<string> characteristics[100];
 
int num_common(int i, int j)
{
  int count = 0;
  vector<string> &v1 = characteristics[i], &v2 = characteristics[j];
  for (int i=0; i<v1.size(); i++)
    for (int j=0; j<v2.size(); j++)
      if (v1[i] == v2[j]) count++;
  return count;
}
 
int main(void)
{
  ifstream fin ("guess.in");
  fin >> N;
  string s;
  for (int i=0; i<N; i++) {
    int K;
    fin >> s >> K;
    for (int j=0; j<K; j++) {
      fin >> s;
      characteristics[i].push_back(s);
    }
  }
    
  int largest = 0;
  for (int i=0; i<N; i++)
    for (int j=i+1; j<N; j++)
      largest = max(largest, num_common(i,j));
 
  ofstream fout ("guess.out");
  fout << largest+1 << "\n";
  return 0;
}