Vanity, which is part of what we now call narcissism, has been recognised as a problem for millenia, represented either as a sin, a target for satire and mockery or as a fatal flaw leading to misery. If it wasn't seen as such a problem it wouldn't have been made one of the Seven Deadly Sins (not to be confused with the other use of vain to mean futile). The term narcissism wasn't used before Freud so vanity covers all aspects of it.
Narcissistic personality disorder — one of several types of personality disorders — is a mental condition in which people have an inflated sense of their own importance, a deep need for excessive attention and admiration, troubled relationships, and a lack of empathy for others.
Narcissism is the pursuit of gratification from vanity or egotistic admiration of one's idealised self image and attributes... Narcissism is a concept in psychoanalytic theory, which was popularly introduced in Sigmund Freud's essay On Narcissism (1914).
In Vanity Fair (which is a good read) some of the characters are at the extreme of both vanity and narcisissism. As they are in the original from Pilgrim's Progress, but much less amusingly.
This is a long-winded (yes, I'm bored) way of saying I don't think it's a modern problem.
Concern about it sounds to me similar to the Is the Internet Eating Teenagers Brains? type of clickbaity headline, or complaints about a lack of empathy. There have always been vain, selfish, anti-social etc people - we all know some - and I suspect that the Internet and its spawnings just make them more salient.
Yes, some people really do love talking about themselves and being the centre of attention. Attention-seeking in this context is what my mum used to call showing off, a great sin. She also said constantly that it's what on the inside that matters which is so not true, especially for teenagers and women generally.
One small example: I've noticed as I've got a bit older that a certain type of man who was quite quiet earlier in life turns into a middle-aged windbag droning on and on about himself and his opinions at any opportunity, not interested in anyone else's opinions or feelings. If someone does manage to get a word in then the man just waits for them to draw breath to bring the conversation back to himself. I've met and been bored by so many of them. They're definitely not part of the selfie generation.
There's an argument to be made for the rise of the Individual as promoted by Thatcher et al at the expense of collectivism and social cohesion in which empathy plays a role but that's a different can of worms.