What "hearing tone" actually means

Tone is a contour — a shape pitch traces over a syllable, not a fixed frequency. A rising tone said by a bass voice and the same rising tone said by a soprano occupy completely different frequency ranges, but they're the same tone, because what matters is the shape (and, in tones like Mandarin's dipping third tone, the relative height compared to the speaker's own range) — not the absolute pitch. Learning to hear tone means learning to listen for shape and relative movement, and specifically learning to ignore absolute pitch as a cue, since it varies by speaker and tells you nothing about which tone was said.

Exercises that actually build this

Minimal-pair drills: two words identical except for tone (mǎi/mài in Mandarin, or a similar pair in your target language) force you to isolate the one variable that matters.
Same-speaker repetition: hearing all four (or six) tones from one consistent voice, back to back, removes speaker-to-speaker pitch variation as a confound while you're still learning the shapes.
Production-then-comparison: saying a tone yourself and immediately comparing it to a reference at the same pitch trains the ear and the mouth together, rather than treating listening and speaking as separate skills learned in sequence.

Why absolute-pitch matching helps here specifically

The second and third exercises above both work better when the reference voice sits at your own pitch. If it doesn't, every comparison you make is contaminated by a register difference that has nothing to do with tone — you end up unconsciously training on "does this sound like a different voice" rather than "does this tone shape match." Removing the register mismatch is exactly what pitch-matched playback (resampling a reference voice into the learner's measured register, preserving contour shape) is for — it isolates the one variable, tone, that's actually meant to be the exercise.