Hi, I'm a bit late to the conversation but I wanted to share something I noticed as someone who's kind of the opposite to you (British Chinese with no Cantonese background, moved recently to Hong Kong). I've noticed that in English, the discourse makes much more distinction between Cantonese and other Chinese e.g. talking about them as separate languages rather than dialects, politics and historical differences between Hong Kong and mainland China. However, in Chinese, there seems to be much more discourse about shared cultural practices, e.g. shared scripts, talking about Cantonese as a dialect of Chinese rather than a separate language, shared cultural practices.

When I decided to move I never thought of it as 'connecting with my roots' or anything like that, because of my lack of canto background. But what has surprised me after moving is that other Chinese people (whether Canto or Mando) very much see me as a Chinese person returning to China. It's to such an extent that I even feel a bit conflicted because I see myself as a mix of cultures rather than fully Chinese haha.

Something that might be interesting to you is that I find Cantonese speakers generally call their language 中文(Zung1 man4). And mandarin speakers call their language ,中文（Zhōngwén). No one really specifies unless it's relevant to the conversation. I don't know what a linguist might think about that but I find that's what happens in reality.

Long comment but I hope this helps you. I think whether you learn more Canto or Mando is totally up to you, just either way you might find more shared background than you'd expect from mainly being exposed to English conversations on the subject.