Get common elements with index from two list C#

LINQ has operations to project a sequence using indexes, but this isn't built into the query expression syntax, so you have to use "regular" extension method calls to start with. After that it's fairly easy, although probably just as simple not using LINQ, to be honest:

var pairs1 = list1.Select((value, index) => new { value, index });
var pairs2 = list2.Select((value, index) => new { value, index });
var matches = from pair1 in pairs1
              join pair2 in pairs2 on pair1.value equals pair2.value
              select new
              {
                  Value = pair1.value,
                  Index1 = pair1.index,
                  Index2 = pair2.index
              };

(You could use from pair2 in pairs2 where pair1.value == pair2.value if you'd prefer...)

Or non-LINQ (using Tuple<,,> for simplicity; other options are feasible):

var results = new List<Tuple<int, int, int>>();
for (int index1 = 0; index1 < list1.Count; index1++)
{
    for (int index2 = 0; index2 < list2.Count; index2++)
    {
        if (list1[index1] == list2[index2])
        {
            results.Add(Tuple.Of(list1[index1], index1, index2);
        }
    }
}

Note that unlike a regular intersection operation, both of these can give you multiple results for the same value - because there can be multiple index pairs. For example, with lists of { 1, 2 } and {2, 2, 0}, you'd have tuples of (value=2,index1=1,index2=0), (value=2,index1=1,index2=1).