Rent-A-Girlfriend's main protagonist, Kazuya Kinoshita, is a bit of a sadsack -- relying on a rent-a-girlfriend service to make himself feel better after being dumped. Throughout the first season, he develops into more of a decisive and intentional person, which leads him to want an actual relationship with his rental girlfriend, Chizuru Mizuhara. But a tiny detail from early in Season 1 may spell disaster for any relationship he chooses to pursue.
His grandmother, Nagomi, mentions that Kazuya is the heir to the Kinoshita family. This means that he's set to possibly inherit money, property or even a family business. This could certainly complicate any long-term relationship he plans on pursuing with Mizuhara, as well as the other girls that surround him.
Kazuya's romantic pursuit of Mizuhara is complicated enough as it is, but the inclusion of him being the family heir will definitely throw a wrench in any long-term plans he may have. Early in Season 1, his grandmother mentions multiple times that he's set to be the family heir, so his choice of partner is important. This explains why Nagomi cares so deeply about Mizuhara, thinking that she's his actual girlfriend. Nagomi wants to know and feel secure that her grandson has chosen a partner that can handle the potential responsibilities thrust upon them by his inheritance.
What exactly Kazuya's set to inherit hasn't yet been made clear in the anime, leaving the fans questioning the possibilities. From the way Nagomi talks about it, it seems like a pretty sizable burden. It could be an inheritance of lots of money, which would definitely complicate things with any partner. It could also be inheriting a family business, which would put Kazuya on a path that he's really not used to. Bearing the weight of a family business means meeting generational expectations and upholding a lifestyle that his parents kept going before him. Either way, his inheritance will definitely play a role in the relationships around him.
If he's set to inherit a family business, or expected to become the next head of the family, Kazuya has a lot of growing up to do before then. That's why Nagomi wants to make sure his girlfriend/wife is also up to the challenge. If Kazuya is left by himself, he may not be able to handle the inheritance thrust upon him. Hence why Nagomi likes Mizuhara -- she's smart, responsible and extremely well-mannered, at least in front of Nagomi. If Mizuhara and Kazuya ended up together, he'd have someone by his side who'd make up for his shortcomings.
This inheritance really has the potential to mess up Kazuya's current lifestyle. His lack of a job, his focus chasing girls -- this will all have to change. Plus, the other girls will definitely start to act differently around him when they realize he's set to inherit something that could potentially change their lives for the better -- especially Mami. She seems to be the kind of person that covets what other people have, so if Kazuya is set to inherit lots of money, for example, then Mami would likely come running to sink her claws into him, securing a financially stable future for herself. For now, as the girls don't know this important fact about Kazuya, so nothing will really change. Only Mizuhara knows anything, and it's clear that it doesn't change her opinion of him, considering she sees herself as simply a rental girlfriend.
It'll be interesting to see what Kazuya's inheritance actually is and what it will mean for his lifestyle. Needless to say, Mami and the other girls that like him will have a new perspective on him once it's revealed. Things will absolutely change for him, whether he's ready or not. So far, only Mizuhara seems up to the task of being by his side with this knowledge, even though it's all pretend. This means that money and success don't mean much to Mizuhara, and if she starts to develop feelings for him, they're going to be genuine. While his inheritance is still up in the air and up for speculation, he'll continue to pursue Mizuhara despite what his family may want.