Given the problem, we aim to find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ satisfying the following conditions for every pair of positive integers $(x, y)$:

1. $x$ and $f(x)$ have the same number of positive divisors.
2. If $x \nmid y$ and $y \nmid x$, then:
$$\gcd(f(x), f(y)) > f(\gcd(x, y))$$

Let's denote $d(x)$ as the number of positive divisors of $x$.

### Analysis

#### Condition (i): Number of Positive Divisors

The first condition implies that $d(x) = d(f(x))$. This suggests that $f(x)$ must be related to the structure of $x$ in a way that the number of divisors remains the same.

#### Condition (ii): GCD Condition

For distinct positive integers $x$ and $y$, if $x \nmid y$ and $y \nmid x$, then the greatest common divisor condition
$$\gcd(f(x), f(y)) > f(\gcd(x, y))$$
suggests that $f$ preserves some multiplicative properties, ensuring that $f$ does not assign smaller values or inappropriate divisibility relationships.

### Solution

Since $f(x)$ and $x$ share the same number of divisors, a natural choice is to let:

$$f(x) = q^{d(x)-1}$$

where $q$ is a fixed prime number. This function $f(x) = q^{d(x)-1}$ has the following properties:
- The number of divisors of $q^{d(x)-1}$ is $d(x)$, aligning with condition (i).
- This choice maintains the strict manipulation of the number of divisors, ensuring multiplicative relationships hold, satisfying condition (ii) when $q$ is a fixed prime.

Therefore, the solution to the problem, where every condition is met, is:

$$\boxed{f(x) = q^{d(x)-1}}$$

This conclusion follows from the necessity to find a function that inherently matches the divisor count while satisfying the inequality requirements imposed by the gcd condition.