Class does matter in that childhood opportunities for the future are so strongly influenced by their background. One of the greatest indicators is maternal education. Much more strongly than the father's. There will be outliers that defy these strong social trends, but these trends are real and the reason for policies such as pupil premium in education.
Class is more muddled and indistinct than it once was. It is much more than money and about culture and aspiration too.
My dad started off in the slums that became post-war council estates. Infact his dad ended up spending the rest of his life in a council flat. My dad passed his 11+ but refused to go to the grammar school, swayed by peer pressure. He started work on the factory floor and worked his way up and studied at night school to improve his opportunities. He progressed up the ranks of management assisted greatly by his combination of intelligence and social skills to adapt to the increasingly privilaged middle classes that he directly worked with, yet maintaining loyalty to his roots, held in great respect by the factory workers who still felt he related to them. By the end of his working life he still firmly held one foot in his working class roots yet was very comfortable in a very middle-middle class social scene. It was that middle class background that I was raised in.
Being able to climb the ranks from the ground up is probably harder now with an emphisis on accademic routes and less on apprenticeships. Access to the top end of the accademic world is not just about a string of A*s, but about being in a school where the staff have the background to see the potential and prepare for less common processes such as entrance interviews.
Financially, middle class is more about comfort rather than pure income. Having a disposable income. Having a safety net so that loss of a salary is not an imminent threat. The less reliant on salary with greater buffers of savings, home ownership, assets and financial products e.g. pension, health insurance, the more middle class. Lower middle class will superficially have many trappings of a middle class lifestyle, but less security behind them such as finance packages to run a fancy modern car, or more vulerability in the mortgage (although mortgages are so strongly influenced by local prices and are a reason why it's now harder to distinguish by home type).
Class is probably a stronger social construct in England due to the history of the feudal system imposed by the Normans, and our lack of gusto at revolution 