It's either. I don't think Navajo doesn't have gendered pronouns or nouns.
06/02/20 This is my first time encountering the word in the course and there's no additional module reading yet, so I'm relying exclusively on the hoverhint to find out what 'Bee' means - which says 'he/she/it' - but upon entering any of those it marks you wrong and corrects you to 'he has'.
Also, the audio on this one needs some equalising as it currently has a huge momentary leap in volume half way through.
Because the course creators have done a ❤❤❤❤❤❤ job making the course. Also, the reason why it's "bee hólǫ́" rather than just "hólǫ́" is because postpositions decline for person and number, much like how prepositions do in Irish Gaelic. I'm not a Navajo expert, but I think you have to use "bee" with "hólǫ́" to specifically say "have", because as you pointed out, "hólǫ́" can also mean "there is/are" and "to exist".
The audio sounds like "ee" to me, not "bee" -- that initial b is difficult to hear, even when I listen for it, and especially with an electronic blip during the second pronunciation.
Note: I mean the audio as available in the "Type what you hear" exercise as of 15 July 2020.
Not the right translation, if you want to understand this word in depth try NavajoWOTD as a supplement to the Duolingo course.
--(quoted)-- Bee is one of a group of words that translates to “with or by means of (it).” Using it takes some practice, but it’s essential knowledge. Here’s an overview.
For reference: Shee would mean “with me”, nee means “with you”, and bee takes its place in the third person “with him/her/it”. These words follow special rules that makes bee (or yee – the other 3rd person form) relatively common.