I only know Blair from a far distance but never had the sense the masses over there felt he particularly "cared about people like me".
UK politics is far less based on the personal charisma of the candidate than the U.S.
even with Boris + Corbyn, you're still voting primarily for the party/policy rather than the person
the emotional connection with voters is much more important for a presidential than a prime ministerial candidate
whereas the ability to manage your own party is more important for a PM (not that it's unimportant for a president, but less directly hands-on)
as far as the legitimacy of Clinton's empathy, I'd guess like most things about him it's complicated
clearly he has a great ability to make people feel like he cares about them and their problems
I'd guess some of it's innate, but at least some of it is having grown up in tougher circumstances than most presidential candidates, and having to claw his way up a bit
Caesar was the probably the greatest political empathizer in history, and he famously grew up in the the Suburra (a slum of ancient Rome) in an apartment building his family owned, and was a self-made man by Roman political elite standards