Ezra eyes go visibly wide at the name 'Skywalker'.

"Lt. Commander Bridger, under General Hera Syndulla's command, with the Rebellion. I do know about a Jedi Knight who was a General in the Clone Wars with that family name? Anakin Skywalker."

That...would fit. For several reasons. Master Kenobi knowing about and being invested in the well being of his former padawan's son made perfect sense. If he's right about Anakin being behind the mask of Vader, all the most reason to want to keep Anakin's son hidden and safe, until he was older.