Angela Merkel wastes no time getting to the point.

Ms. Merkel, the chancellor of Germany, projects a managerial efficiency at summit meetings with a style that borders on flavorless. She enters briefings rapidly, steps purposefully onto a dais, and settles into a modern, high-backed office chair. Behind her is a giant map of Europe dotted with yellow stars representing the bloc’s 28 capital cities, and underscoring Germany’s decades-long mission to further integration. There is a hush as journalists parse the words of the leader of the group’s most powerful country. The atmosphere can occasionally turn reverent, as when a journalist serenaded Ms. Merkel for her 60th birthday. But her power at home is now being challenged over her policy on migration, potentially complicating her role as Europe’s de facto leader.