In asymmetric cryptography, is it possible to ensure that public key derived from a given private key is unique, no matter the algorithm?
You cannot speculate on unknown algorithms.
With RSA, you can make several public keys which are all functionally equivalent, in that they all correspond to the same private keys. Namely, if the public modulus n is the product of p and q, and the public exponent is e, then e'=e+k*lcm(p-1,q-1) will be another valid public exponent for the same modulus, and corresponding to the same private key, for any integer k ("lcm" is the Least Common Multiple). This allows you to compute an infinity of distinct public keys for one private key. (But it is not recommended at all to make more than one of these public keys public: combining two such keys reveals the private key quite easily.)
If you want unicity of public key, for some notion of unicity, then you have to arrange that yourself.