RSA public key same across all clients or different? I understand that public key is a pair of $(e, N)$ where $N$ is product of two primes and $e$ is encryption key.

I understand that $N$ is usually unique for an entity. For example, Facebook, Google, etc. have $1028$ or $2048$ bit $N$ that they publish.

What about $e$? Is it also unique or do these entity give different $e$ to different clients/customers? Does it changes from one session to other for same client?
The full public key is usually published, not just $N$. The public key consists of $N$ and $e$ + meta data (ASN.1 encoding, object identifier for RSA, etc.). How the public key is encoded depends of course to the protocol.

If it is shared for e.g. transport security it is usually part of a certificate that also contains subject, issuer, validity period, key usuage etc. and of course the signature of the issuer over the certificate. This makes it possible not just to retrieve the public key but also to find out if the key should be trusted and how the key should be used.

Generally the public key uses the Fourth prime of Fermat (F4) as public exponent $e$. This makes public key operations somewhat faster and it allows for easy key pair generation. Some cryptographic libraries or runtimes restrict the size or value of $e$, so choosing a different value is not recommended. Still, most libraries will accept an $e$ that is, for instance, smaller than 4 bytes (i.e. $2^{32}$ for unsigned or $2^{31}$ for signed integers). Using F4 is generally not a requirement in protocols. That said, using anything different may cause compatibility issues.

Having a different $e$ doesn't affect the security of RSA, as long as it is relative prime to $φ(pq)$.



Notes:


Using 1024 bit RSA is not recommended anymore. Using a 2048 bit or higher key size is highly recommended.
$N$ directly depends on the two (or more) random primes used to generate the key pair so $N$ is always fully unique, making the public key fully unique - regardless of $e$. A hash over $N$ is generally used as key ID or fingerprint of the key pair. $N$ is part of both the public- and the private key.