By all accounts, the two women have a complicated dynamic, and they coexist with little overlap in their roles. But they have not hosted a joint initiative carried out solely between their staffs since Mrs. Trump moved full time to the White House last year after spending the first months of her husband’s administration with her son in New York. They have rarely appeared together. And they clearly see their roles differently.

Melania Trump, 48, prefers to stay deeply private. Ivanka Trump, 37, has sought from the earliest days of the Trump administration to define her role as an adviser and policy maker, staking out a claim to an office in the West Wing, where the president and his staff are, after early reports of a Trump family office potentially being set up in the East Wing, the traditional domain of the first lady.

And while expanding her own presence in the White House, Ms. Trump has at times, intentionally or not, defined her stepmother’s role in more limited terms.

Friends say she has noticeably bristled when asked questions that she saw as traditionally in line with a first lady’s responsibilities — among friends, she has dismissed queries about whether she would be involved in White House preservation efforts, and has made it clear that she was in the White House to work on meaty policy issues, a move some allies say was out of deference to Melania Trump.

Daughter and wife both have influence with the president but they exercise it differently. Earlier this year, the first lady, who has made it a point to say she is willing to disagree with her husband, spoke publicly about her discomfort with the administration’s policy of separating children from their parents after illegal border crossings — and made several trips to the area.