I think it's our Kurama. Why ? When a tail beast dies. Their original chakra gathers & forms their body again and their soul returns into it . Three tail was the same "ego". Before & After dying.

Baryon mode burned his original chakra. So Kurama thought His soul will go to after life. " Until we meet again Naruto... Don't come too soon -> implying he was going to afterlife. " Let's stay together in the afterlife,, Naruto suggested the same idea that Kurama's soul was going to after life.

But Kurama didn't realized that his chakra was passed to Boruto & Hima via Naruto. Jinjuriki's child gets a very small amount of tail beast chakra.... it's inconsequential in terms of both power & it usually serves no purpose. As long as The tail beast didn't burn his original chakra to nothingness using nuclear physics.

Kurama's soul (soul or consciousness whatever Kurama thought will go to pure land instead ) didn't go to afterlife but went inside Hima. & Kurama began passively pulling in nature chakra. He was maybe unconscious for a while (it’s possible happened in naruto vs sasuke final battle) and only woke up after he gained back like 10% of his original power. He woke up to find Naruto was gone. He decided to stay quiet and gain back his original power completely before confronting kawaki. Because Of Kawaki he kept everyone in dark, Kawaki might have tried something. Thats why he said "to think ten tail would end up sniffing me out." !!

As i said it was a slow process so post baryon mode naruto & sasuke didn't sensed anything inside hima ! But now kurama has gained back enough streng to be sensed. Plus Jura (& Daemon) might just be a greater sensor than Naruto & Sasuke.

Boruto-> Momo repelled it like how kurama repelled Maa & Paa to connect to Naruto during Sage mode training. Boruto is like completely changed genetically anyway. He is otsutsuki now not human. Or simply Momo feared kurama chakra will rather help Boruto fight momo (& remain in control / not run out of chakra) than Him so he repelled it.

i don't see how a entire new soul will be formed. Hagoromo used creation of all things to give them consciousness. & even when split (in Minato & Naruto,, the two variants shared past memories) While kinkaku and ginkaku 's varient didn't have consciousness at all.... So since the nine tail inside hima has one, i believe its our Kurama. I think just like kinkaku & Ginkaku variant, hima just had the chakra without consciousness before the baryon mode happened & Kurama's soul went into her.

Also if it was a new being, why did he remained quiet? It’s every living being’s instinct to be freee & not cofined…. & let’s not forget there is no seal… He should at-least be curious like ten tails are & tried talking to hima…

Kurama casually began talking to her after he was “sniffed out by ten tail” & thought its time… but when he saw Hima is shocked & couldn’t understand what’s happening…. He asked her “do you know who i am ?” or “ do you understand what i am? “ …. Kurama knows everyone has heard of him but hasn’t seen him or met him… Only Kawaki did. Plus he looks different now.

“little girl” thats how kurama talks. Like an old man. He knows her name is Hima.