I like the idea of the attacks being related to the player's status as Dragonborn, and also agree that this could have been hinted at a little better, say in the assassination order found on the Justiciars who attack.
However, I also feel that the Thalmor's tendency to attack you on sight, with little to no provocation from the player themselves either in the past or in the moment, reflects the nature of the world we're playing in and the idea that the Thalmor represent. They're the Nazi's. The Gestapo. The special military/law enforcement service who gets to arrest, imprison, or execute anyone they wish under a single, difficult to disprove charge that can be applied to anyone they wish to see eliminated for any reason whatsoever. Whether it's because you're the Dragonborn, or because you've embarrassed/threatened/challenged the Aldmeri Dominion, or accidentally put on brown shoes with a black belt. The Thalmor want you dead because reasons and they said so, so an order is put out under the blanket "Talos worshiper" accusation.