What happens if an RSA key pair has identical public and private exponents? Rather, is it possible for big prime numbers?

Classroom examples usually involve smaller primes, so for example if you are given a prime number pair $p = 3$, $q = 13$ you would get $n = 39$ and $e = d = 5$, making encryption and decryption the same for a message $x$ or ciphertext $x$.

This can be a big problem, e.g. if a server sends you cipher $x$, you can decrypt it and get the message, then you encrypt the cipher $x$ itself and if you get the message again then you know the public key is the same as the private key.

Is this possible in industry?
No, the public and private exponents will never be the same for real (that is, not toy) RSA keys.

The public exponent is almost always be deliberately chosen as a small value (with 65537, 3 and 17 being the most popular choices).  In contrast, the private exponent will always be a huge value; always at least $(p-1)/e$ (where $p$ is the larger prime factor of the modulus, and $e$ is the public exponent), and will generally (almost always) be much larger than that.  This implies that if you have a 1024-bit RSA key, and a public exponent of 65537, then the private exponent will be at least 495 bits long.

Even if you (for some unknown reason) select a random large public exponent, then it is still extremely unlikely.  If we look into the conditions that allow $d = e$, we see that both of the following must hold:

$e^2 \bmod (p-1) = 1$

$e^2 \bmod (q-1) = 1$

Having even one of $e^2 \bmod (p-1)$ and $e^2 \bmod (q-1)$ to be 1 is extremely unlikely; having both conditions happen is not something to worry about.