She's happy to receive expensive gifts from her biggest fan, a ostensibly wealthy middle aged Japanese man, yet is a bit ambivalent about her relationship with him and whether it should continue despite the gifts he sends which please her.

I don't know what kind of a "relationship" they have beyond him being a fan and sending her gifts, and her happily receiving them, but if that's the extent of the relationship then I don't see a problem. Any other aspects of the relationship should be discussed and decided on between both parties. There may be a sense of expectation involved - that by accepting the gifts, this girl is tacitly granting this man a certain entitlement - to who knows what. But I think it's unfortunate if our conception of "gifting" has become that tainted. Gifts should be given without expectation of reciprocation. If you want something from a person, you can discuss that with them, and if they're not willing to give it, you may be able to come to a compromise - but a gift given without explicit conditions for acceptance should have no strings attached.

Anyway, she's already giving him something I suspect he enjoys - the opportunity to admire her through her videos/photos/whatever it is she's become famous for.

In the documentary, the family reveals that they've all met this man, and that he's very polite and quiet, "Like most Japanese men", and that he considers himself to be more of a "friend" than a "fan". Just how many times they've met him in person is left unclear, but they definitely have some sort of relationship with him. The mutual sincerity of that relationship is also unclear, but we do know that they've met in person, and we see her opening up his birthday gift to her, an expensive Fender bass guitar that he sent her. We then see her emailing him to express her gratitude for the gift.

It seems as if the family likes or at least tolerates this man, as he's very polite and it's understood that idols in Japan don't have boyfriends anyway, so he knows that his crush won't lead anywhere physical. But the culture clash is apparent by the ambivalence that Beckii shows towards him. While this would all be quite normal to a native Japanese idol and her family, to Beckii and her family it's a strange, perhaps somewhat creepy, cultural experience which they accept out of equal parts ambition and cultural tolerance and politeness.