The organisations are very much separate. Football clubs in the UK and UEFA have stringent financial rules and an open books policy in which all spending has to be within the range of the clubs operating financials. The Premier League has an 85% of turnover rule and UEFA is moving to 70% of turnover, though they're trying to get more aligned.
This means that it doesn't matter a jot how rich your owners are, you can't spend more than the club brings in from its own enterprises. There are loopholes which have started being closed, such as Chelsea inexplicable selling themselves their ladies team for several hundred million dollars, or spreading the cost of the transfer fee to buy a player over the length of a contract (average for new signings is around 4/5 years - Chelsea started buying players and putting them on 8 year contracts which just is not done in European - or world - football), and Manchester City are under investigation for 116 charges such as making up companies for their owners to funnel more money to them through sponsorship and could lead to them being stripped of titles, but in the main, it's a pretty tight ship. Teams routinely get point deductions and fines for not complying.
So Henry could want to move all the money across he likes. It isn't happening. It's by far the most annoying idiocy I hear from fans. A 2 minute google will tell you all you need to know. But fans prefer to be emotional and irrational about all aspects of fanhood. Proud of it even (not aiming this at anyone in particular, just the standard level of fan engagement in learning about actual facts). They are separate entities and while Henry's time may well be divided (he's not involved day-to-day with Liverpool, or anywhere near it), his money and assets are not.
As an aside, despite bringing a lot of success to Liverpool, there was (tailed off a bit now after another league title last season) a significant and vocal FSG OUT contingent of the fan group who (despite the above mentioned constraints) embarrassingly complained constantly that FSG didn't spend enough. They have been for the most part (there have been several huge errors) excellent owners for Liverpool. What they have in place at Liverpool that the Sox don't, is an absolute top notch management team. You will often hear 'best in class' said about Liverpool and how they do their business. They are run with ruthless efficiency and rarely take a wrong step in their transfer dealings. If the Sox had the same sort of brain power working at the top of their organisation, Henry's wet dream of having a finely tuned machine at half the price may be possible.