That's essentially what a Vocoder offers. Combined with ARPs - you could end up with single-finger chords.

https://www.yamahasynth.com/modx-category/moessieurs-monday-the-modx-vocoder

https://www.yamahasynth.com/yamaha-synth-rss/moessieurs-monday-the-modx-vocoder-part-ii

https://www.youtube.com/watch?v=pcLIl304dIk

https://www.youtube.com/watch?v=cULDwmhfPNE (this one is more single-note, not intervals - but here to show the kind of sound you could get out of chords if you played the same thing with multi-note)

The distinction may be that a vocal harmonizer will traditionally try its best to match the sound of the incoming sound to the outgoing sound. Meaning - it duplicates your voice (or instrument) into unison or harmonic intervals while retaining the original sound/timbre. A vocoder does not do this - the outgoing sound is characteristic of the formant of the incoming sound (the "vowel" sound) but the timbre of the outgoing sound is different. Often "robotic" sounding.

There is no harmonizer that will reproduce the same incoming signal and pitch-shift it in real-time. The vocoder, as described, is as close as you get.

If you want to harmonize an incoming vocal or instrument - it would not be reacting to what you play on the keyboard (and harmonizing with the notes you play) - but you could use the insertion effect: "Pitch Change" under the "Misc" category. Each of these effects gives you up to two different notes (per insertion effect) you can "harmonize" with the incoming vocal at fixed intervals. Because insertion effects can be controlled/offset with controllers, you could change the chord quality so all the notes that are pitch shifted could themselves be offset in pitch. This would entail maybe using the A.SW1 and A.SW2 buttons or somehow otherwise programming a way to control the pitch offsets. This could create a semi-manually-controlled harmonizer if you could keep up with the button pressing or knob turning. Since there are two insertion effects - you could have up to 4 shifted note superimposed (if you wish) with your original voice/instrument.

EDIT: To clarify - the pitch effect, as you'll notice throughout the keyboard, does not allow for all parameters to be offset. One you cannot offset externally is the pitch adjust. Therefore, if you want to harmonize different notes and control these with A.SW1/A.SW2 (for example) - you'll need to setup two "Pitch Change" effects - which gives you 4 different notes. Then assign the output level for each note to an offset (as these can be destinations). This will enable you to selectively turn "on" and "off" different intervals which could be useful - although not as complete as being able to "dial in" the offsets for each of the 4 "pitch change" notes.

It's sort of possible to link keyboard keys to offsets by making certain keys output at different volumes then using envelope follower to control (edit: ) output levels to turn on/off different pitches. That can get convoluted - but about the only way considering note value is not a controller source for more direct control.