Twofold recursion, over both arrays - see the comments:

// recursives
void find_matches(int *arr, int *bee, int *matches, int current_position_in_arr);
void find_matches_in_bee(int *arr, int *bee, int *matches, int current_position_in_arr, int current_position_in_bee);

// wrapper
void find_matches(int *arr, int *bee, int *matches) {
    find_matches(arr, bee, matches, 0);
}

// outer loop : iterate over 'arr'
void find_matches(int *arr, int *bee, int *matches, int current_position_in_arr) {
    // check where arr[current_position_in_arr] is present in bee
    find_matches_in_bee(arr, bee, matches, current_position_in_arr, 0);

    // "next iteration of loop" - we check the next element in arr
    if (current_position_in_arr + 1 < length) {
        find_matches(arr, bee, matches, current_position_in_arr + 1);
    }
}

// inner loop : iterate over 'bee'
void find_matches_in_bee(int *arr, int *bee, int *matches, int current_position_in_arr, int current_position_in_bee) {
    // do your business magic
    if (arr[current_position_in_arr] == bee[current_position_in_bee]) {
       ....
    }

    // "next iteration of loop" - we check the next element in bee
    if (current_position_in_bee + 1 < length) {
        find_matches_in_bee(arr, bee, matches, current_position_in_arr, current_position_in_bee + 1);
    }
}

Invoke the same way as before:

find_matches(arr, bee, matches);

The lesson here is that you can replace stuff like:

int *array;

for (int i = 0; i < LEN; ++i) {
  f(array[i]);
}

with

void f(int *array) {
  f_inner(array, 0);
}

void f_inner(int *array, int position) {
  // business logic on array[position]

  // iteration step
  if (position + 1 < LEN) {
    f_inner(array, position + 1);
  }
}

like this :

#include <stdio.h>
#include <stdlib.h>
#include <math.h>
          //  0  1  2  3  4  5  6  7
int arr[] = { 3, 6, 1, 2, 6, 7, 8, 4};
int bee[] = { 6, 8, 1, 4, 2, 6, 3, 7};
int matches[120] = {0};
int length1 = 8;

void find_matches(int *arr, int *bee, int *matches, int i)
{
    if (i == length1)
        return ;
    for (int j = 0; j < length1; ++j)
        if (arr[i]==bee[j])
            matches[i] = j;
    printf("%d\n", matches[i]);
    find_matches(arr, bee, matches, i + 1);
}

int main()
{
    find_matches(arr, bee, matches, 0);
    return 0;
}

Your code is clean and clear as presented in nested loops. Why would you want to delibrately convert it to recursion? If you really want, how about

#include <stdio.h>
#include <stdlib.h>
#include <math.h>
          //  0  1  2  3  4  5  6  7
int arr[] = { 3, 6, 1, 2, 6, 7, 8, 4};
int bee[] = { 6, 8, 1, 4, 2, 6, 3, 7};
int i = 0;
int j = 0;
int matches[120] = {0};
int length1 = 8;

// iterate over bee
void match(int i, int j)
{
    if (j < 0)
        return;
    if (arr[i] == bee[j]) {
        matches[i] = j;
        return;
    }
    match(i, j - 1);
}

// iterate over arr
void fit(int i)
{
    if (i > 7)
        return;
    match(i, 7);
    fit(i + 1);
}

void find_matches_recursively()
{
    fit(0);
    for (int z = 0; z < 8; z++)
        printf("%d\n", matches[z]);        
}

int main()
{
    find_matches_recursively();
}

Output

6
5
2
4
5
7
1
3

which is the same as in your code.

Note: You have duplicated values in your bee[8] (value 6 in position 0 and position 5). Your iteration loops over all values and records the last match. My recursion searches from the tail and returns immediately when it finds the first match. The effect is the same.