For anyone that finds this question looking for how to access custom properties in ASP.NET Core 2.1 - it's much easier: You'll have a UserManager, e.g. in _LoginPartial.cshtml, and then you can simply do (assuming "ScreenName" is a property that you have added to your own AppUser which inherits from IdentityUser):
@using Microsoft.AspNetCore.Identity
@using <namespaceWhereYouHaveYourAppUser>
@inject SignInManager<AppUser> SignInManager
@inject UserManager<AppUser> UserManager
@if (SignInManager.IsSignedIn(User)) {
<form asp-area="Identity" asp-page="/Account/Logout" asp-route-returnUrl="@Url.Action("Index", "Home", new { area = "" })"
method="post" id="logoutForm"
class="form-inline my-2 my-lg-0">
<ul class="nav navbar-nav ml-auto">
<li class="nav-item">
<a class="nav-link" asp-area="Identity" asp-page="/Account/Manage/Index" title="Manage">
Hello @((await UserManager.GetUserAsync(User)).ScreenName)!
<!-- Original code, shows Email-Address: @UserManager.GetUserName(User)! -->
</a>
</li>
<li class="nav-item">
<button type="submit" class="btn btn-link nav-item navbar-link nav-link">Logout</button>
</li>
</ul>
</form>
} else {
<ul class="navbar-nav ml-auto">
<li class="nav-item"><a class="nav-link" asp-area="Identity" asp-page="/Account/Register">Register</a></li>
<li class="nav-item"><a class="nav-link" asp-area="Identity" asp-page="/Account/Login">Login</a></li>
</ul>
}