I work in a field with HIPAA protections.

Definitely need to be careful then.

is this kind of software considered “AI”?

The best voice recognition is based on AI — yes.

Before AI, voice recognition existed but it was generally pretty shit and really struggled with accents, low quality microphones, background noise, people saying things that don't strictly make sense. E.g. if you say "We’ll burn that bridge when we get to it." a good AI might replace "burn" with the word "cross"... it will at least have the capability to do that, wether or not it will would depend on your settings - is "accuracy" about what someone said or what someone actually meant? That's configurable in the best systems.

Does the tech learn to recognize my voice better over time?

Old software did. These days systems work so well that would just add cost with zero benefit. Good speech recognition will understand your speech perfectly as long as your microphone is decent and "learning" wouldn't help much with that one potential problem area.

Some speech systems do learn in order to recognise/identify people (for example, a voice assistant might use it to figure out who "me" is in a command like "remind me to do get milk when I get to the shops". And a good transcription service will recognise different people talking in a single recording, and provide an appropriately annotated transcript. That's about the extent of "recognising" your voice, it doesn't generally learn from you over time.

Is this all a black box?

Kinda yeah. The researchers paid a huge number of people in third world countries to compare recordings to transcriptions, and make a "correct / incorrect" judgement call. Then fed all of that, and a whole bunch of other things (it's believed every YouTube video ever uploaded might have been involved...) into a very complex model.

Tweaks are made but it's just too much data (OpenAI says they used 680,000 hours of audio) to fully get your head around all of it. A bit like trying to understand how the human brain recognises speech — we have a broad idea but don't really know.

Does it use my recordings to learn to understand other’s voices? How can I take precautions such that no one except me hears the things I transcribe?

Check the privacy statement for the service. They might, for example, send your recordings to be assessed for accuracy by employees/subcontractors. AFAIK (not a lawyer) that would be a breach of HIPAA.

AFAIK some Apple speech recognition features are HIPAA compliant. Look that up to verify it but in general iPhones and Macs Apple have AI speech processing hardware on the device allowing fully local processing... but not all features are done locally and in some cases they may transmit "anonymised" (useless if you speak someone's name...) speech to employees/contractors to improve the software. That can be disabled in settings.

Amazon and OpenAI do everything in the cloud but have fully HIPAA compliant versions of their services (I assume those are not cheap...)

You could try open source models — I don't know how good they are in practice.