What angular version are you using?

Have you checked, if user is really written?

Here would be some changes I would try without further context:

Set changeDetection

@Component({
    selector: 'app-header',
    standalone: true,
    imports: [RouterModule, NgIf],
    templateUrl: './header.component.html',
    styleUrl: './header.component.css',
    changeDetection: ChangeDetectionStrategy.OnPush, //Add changeDetection
})

Use @if instead of *ngIf since it's deprecated with angular 20.

@if (user()) {
    <span>Olá, {{ user().name }}</span>
} @else {
    <button class="button" routerLink="/login">Entrar</button>
}