Nothing racist in there mate so no need to worry about that
The first point I'd make is that it should only be up to the individual themselves to decide what their nationality or heritage is. What gives anyone else the right to decide whether someone is British or not without actually knowing that person? It's why I hate hearing the phrase - which has been said to me personally - 'go back home', 'go back to your own country'.
To answer your questions (!), one of my parents was born in India and the other in Kenya (you'll find a lot of Indian people of my age/generation has a parent/parents that were born in Africa; there was a lot of work and trade available in Africa in the 60s/70s so many Indian people migrated, albeit temporarily for some, to places like Kenya, South Africa, Uganda; there's big Indian populations in places like Nairobi, Durban, Kampala for example...). Anyway, I identify as British Asian/British Indian if it needs to be more specific. So yes, while I was born in this country and proud to be so, my heritage is Indian and I will never ever forget that, ignore that or better still, never let someone else choose that or tell me where I am from or what I am.
I would let my kids choose how they want to identify when it comes to nationality or heritage; if they choose to identify as British only, then so be it. If they choose to identify as British Asian/British Indian, again so be it. As long as they recognise where their parents/grandparents/great grandparents and ancestors were from. It's something I would never want them to lose sight of.
I try not to pay too much attention to white British people who say 'we're becoming a minority in our own country'. The UK is still predominantly white (by roughly 60-70% if I remember correctly? Maybe higher). I don't see that changing, if at all, for several generations so they're not even going to live to see the effects of multi-culturalism and even bigger diversity of the land here...