I found a link that helped me to solve this using normals. To solve this, first you calculate the normals of each point of the triangle that the Raycast hitted:

var normalHittedTriangle0 = normals[triangles[hit.triangleIndex * 3 + 0]];
var normalHittedTriangle1 = normals[triangles[hit.triangleIndex * 3 + 1]];
var normalHittedTriangle2 = normals[triangles[hit.triangleIndex * 3 + 2]];

And them the normal of the face of the cube:

var hittedTriangleNormal = ((normalHittedTriangle0 + normalHittedTriangle1 + normalHittedTriangle2) / 3);

And them you loop through all the triangles in the Cube. Get the normal of each point of the triangle, get the normal of the whole triangle and check if it match with the normal of the triangle that the Raycast hitted, if it matched add it to a list.

List<int> trianglesToRemove = new List<int>();
 
for (int i = 0; i < triangles.Length; i += 3)
{
        var p0 = vertices[triangles[i + 0]];
        var p1 = vertices[triangles[i + 1]];
        var p2 = vertices[triangles[i + 2]];
 
        p0 = normals[triangles[i + 0]];
        p1 = normals[triangles[i + 1]];
        p2 = normals[triangles[i + 2]];
 
        if (((p0 + p1 + p2) / 3) == hittedTriangleNormal)
        {
                trianglesToRemove.Add(triangles[i + 0]);
                trianglesToRemove.Add(triangles[i + 1]);
                trianglesToRemove.Add(triangles[i + 2]);
        }
}

Loop through the triangles of the Cube by a step of 3, find the 2 triangles that match with the list and then if it matched, remove it from the triangles of the Cube.

for (int i = 0; i < triangles.Length; i+=3)
{
        bool v1 = (triangles[i + 0] == trianglesToRemove[0]);
        bool v2 = (triangles[i + 1] == trianglesToRemove[1]);
        bool v3 = (triangles[i + 2] == trianglesToRemove[2]);
 
        if ((v1 && v2 && v3))
        {
                triangles.RemoveRange(i, trianglesToRemove.Count);
        }
}

All the code:

if (Input.GetMouseButtonUp(0))
{
        RaycastHit hit;
 
        var camera = Camera.main;
 
        if (!Physics.Raycast (camera.ScreenPointToRay(Input.mousePosition), out hit))
        {
           return;
        }
 
        var meshCollider = hit.collider as MeshCollider;
 
        if (meshCollider == null || meshCollider.sharedMesh == null)
           return;
 
        var mesh = meshCollider.sharedMesh;
        var triangles = mesh.triangles;
        var normals = mesh.normals;
        Transform objectPosition = hit.transform;
 
        var normalHittedTriangle0 = normals[triangles[hit.triangleIndex * 3 + 0]];
        var normalHittedTriangle1 = normals[triangles[hit.triangleIndex * 3 + 1]];
        var normalHittedTriangle2 = normals[triangles[hit.triangleIndex * 3 + 2]];
 
        var hittedTriangleNormal = ((normalHittedTriangle0 + normalHittedTriangle1 + normalHittedTriangle2) / 3);
 
        List<int> trianglesToRemove = new List<int>();
 
        for (int i = 0; i < triangles.Length; i += 3)
        {
            var p0 = vertices[triangles[i + 0]];
            var p1 = vertices[triangles[i + 1]];
            var p2 = vertices[triangles[i + 2]];
 
            p0 = normals[triangles[i + 0]];
            p1 = normals[triangles[i + 1]];
            p2 = normals[triangles[i + 2]];
 
            if (((p0 + p1 + p2) / 3) == hittedTriangleNormal)
            {
                trianglesToRemove.Add(triangles[i + 0]);
                trianglesToRemove.Add(triangles[i + 1]);
                trianglesToRemove.Add(triangles[i + 2]);
            }
        }

        GameObject newCube = new GameObject();
        newCube.transform.position = objectPosition;

        newCube.AddComponent<MeshFilter>();
        newCube.AddComponent<MeshRenderer>();
 
        for (int i = 0; i < triangles.Length; i+=3)
        {
            bool v1 = (triangles[i + 0] == trianglesToRemove[0]);
            bool v2 = (triangles[i + 1] == trianglesToRemove[1]);
            bool v3 = (triangles[i + 2] == trianglesToRemove[2]);
 
            if ((v1 && v2 && v3))
            {
                triangles.RemoveRange(i, trianglesToRemove.Count);
            }
        }
 
        mesh.Clear();
        mesh.vertices = vertices;
        mesh.triangles = triangles;
        newCube.GetComponent<MeshFilter>().mesh = mesh;
}