I would have more issue if it was Abrams who was suggesting that he was such in the first place, but since it's really others who are doing that I think the comparisons are pretty pointless.
A lot have people have been compared to Spielberg over the years but ultimately he is rather unique and it's always an unwise comparison. At best, he belongs in the company of Lucas, Ford Coppola etc and those directors that emerged in the 1970s and then went in different directions, rather than the directors of today.
Spielberg has his flaws and I'm not in awe of his films as I used to be because his films are starting to vary more widely in worthiness - but he is relatively incomparable as a director of pure cinema.
Abrams is fine at what he does, but I wouldn't call him a successor.
'If the Apocalypse starts, beep me!' - Buffy Summers
'The sky's the limit.....' Jean-Luc Picard, 'All Good Things'
courtesy of Saquist