I think this would work out a whole lot more expensive than annual holidays, and not be half so enjoyable.
I agree with what others have said about the duplicated costs and inefficiencies of maintaining two households, and also the emotional aspect of living (and, especially, growing up) with a foot in both countries.
Living here and there also creates so many bureaucratic complications about 'residency' and your entitlements/responsibilities in each respective country. Regardless of whether you're a citizen or not, defining whether you're 'resident' somewhere is a PITA, because every government agency, service provider and commercial company tends to define 'resident' on its own terms. This means you get into endless complications with tax, healthcare, insurance, drivers licences, registering with the local authorities, education, banks, mortgages... almost everything official becomes a headache.
I lived in the US on a non-resident visa for several years: at one point, I was simultaneously non-resident for immigration purposes, but resident for tax purposes, but non-resident for education (had to pay overseas fees), but resident for state driver's license and insurance, but non-resident for banking, but resident for private healthcare, but non-resident for any public healthcare or benefits... and the list continues. But I wasn't counted as residing in the UK, either. When we moved back, and I needed to access pregnancy care, they were scrupulous about checking we were establishing residency and not just nipping back for some free healthcare.
I think it would be a fun way to live if you were super-rich and had people to deal with this shit, but mostly it just ends up feeling like you don't particularly live anywhere.