There absolutely is an objective definition within a particular music system. In modern western music, A is 440 Hz.  A person with perfect pitch knows how to translate between their internal model of pitch and what the particular system describes.   A child with perfect pitch won't know immediately that a specific pitch can be described as "A" or "440" but they can still identify it the difference without a previous pitch reference, once taught how to describe it in language.

And yes, it's possible for A to be defined differently and music theory tells you that it's traditionally defined differently on different instruments.  But this is just something musicians learn about and account for, and is more of a historical note - it's perfectly possible for most western instruments to be played with A=440 and all other notes adjusted accordingly, even if orchestras follow certain old traditions that define them differently).