KENNEBUNKPORT, Maine — Jeb Bush, who has a longtime relationship with this seaside town where generations of Bushes have vacationed, is having a house built for him at the family compound on Walker’s Point, with a wraparound porch and expansive views of the Atlantic Ocean.

The home, on a 1.3-acre site assessed by the town at $1.4 million, was initiated for him by his mother and father, who have tried to preserve the family ties to the picturesque property.

But as he tries to appeal to middle-class Americans in his likely Republican presidential campaign — and distinguish himself as his own man, distinct from the legacies of his father and brother — having a vacation home erected on a spit of land in coastal Maine could be a vivid reminder of the complications facing his campaign.

Already, his opponents are emphasizing his last name — and the old-money New England clan that helped build the Republican Party establishment — as a way to highlight a brand that has fallen out of favor with the newly empowered, anti-elite activists who may have an outsized voice in choosing the next Republican nominee.

At 3,000 square feet, the four-bedroom, four-and-a-half bathroom home, with shingle siding and gabled roof, will be the second-largest home at the compound when completed in July. The construction cost of the house, referred to as a cottage in planning documents, is estimated at $600,000....