Most people only have one mobile device, a smartphone. Those that have more than one usually still have those with matched OSes. No-one cares about "looking like the OS", but people do care about looking (and, more importantly,
behaving) like all the other apps they use.
And as far as manuals and customer support, what you're saying is that you can't afford to do cross-platform properly, and so you're cutting corners. Which is fine if it's stated explicitly upfront, and having an app that behaves weirdly (for a given platform) is better than no app, but please don't insult your users' intelligence by presenting that as some kind of feature.