Is RSA signature as strong as the RSA encryption?
Breaking RSAES-OAEP encryption for generic hash functions can't be much easier than computing odd roots modulo a large composite of unknown large prime factorization.  (Likewise RSA-KEM, which is much simpler.)

Forging RSASSA-PSS signatures for generic hash functions can't be much easier than computing odd roots modulo a large composite of unknown large prime factorization.  (Likewise RSA-FDH, which is much simpler.)

(Forging Rabin-Williams signatures—which involves square roots, not odd roots—for generic hash functions is as difficult as factoring a large composite of unknown large prime factorization.)

This is the story for the individual cryptosystems RSAES-OAEP under a key independent of RSASSA-PSS under another key.  The security of a composite cryptosystem (RSAES-OAEP, RSASSA-PSS) under the same key, in which the adversary can witness decryptions and signatures of arbitrary messages, requires its own analysis: security of RSAES-OAEP and RSASSA-PSS individually does not imply security of the composite scheme.

The standard pathological example, in the scheme I like to call do you even RSA, bro‽, is that if I want to decrypt a ciphertext $c \equiv m^e \pmod n$, and if I can ask you to sign the message $c$, then your signature is $s \equiv c^d \equiv m^{ed} \equiv m \pmod n$, so you just unwittingly decrypted $c$ for me.

Are there attacks on (RSAES-OAEP, RSASSA-PSS) under a single key?  Unlikely, but I don't have a handy citation for a reduction of security to the RSA problem.