Double it or triple it? Doesn't matter

$2M$ and $3M$ have the same number of divisors. Then in the prime factorization of $M,$ $2$ and $3$ must have the same exponent.

Case 1 : $M=2^0 \times 3^0 \times m,$ where $m$ has 6 positive divisors. Then $M$ has 6 positive divisors.

Case 2 : $M=2^1 \times 3^1 \times m,$ where $m$ has 2 positive divisors. Then $M$ has 8 positive divisors.

Case 3 : $M=2^2 \times 3^2.$ Then $M$ has 9 positive divisors.

The exponents on $2$ and $3$ cannot get any lower or higher, so these are the only possibilities. Therefore 7 positive divisors is impossible.