How to assert that two list contains elements with the same public properties in NUnit?

REWORKED ANSWER

There is a CollectionAssert.AreEqual(IEnumerable, IEnumerable, IComparer) overload to assert that two collections contain the same objects in the same order, using an IComparer implementation to check the object equivalence.

In the scenario described above, the order is not important. However, to sufficiently handle also the situation where there are multiple equivalent objects in the two collections, it becomes necessary to first order the objects in each collection and use one-by-one comparison to ensure that also the number of equivalent objects are the same in the two collections.

Enumerable.OrderBy provides an overload that takes an IComparer<T> argument. To ensure that the two collections are sorted in the same order, it is more or less required that the types of the identifying properties implement IComparable. Here is an example of a comparer class that implements both the IComparer and IComparer<Foo> interfaces, and where it is assumed that Bar takes precedence when ordering:

public class FooComparer : IComparer, IComparer<Foo>
{
    public int Compare(object x, object y)
    {
        var lhs = x as Foo;
        var rhs = y as Foo;
        if (lhs == null || rhs == null) throw new InvalidOperationException();
        return Compare(lhs, rhs);
    }

    public int Compare(Foo x, Foo y)
    {
        int temp;
        return (temp = x.Bar.CompareTo(y.Bar)) != 0 ? temp : x.Bar2.CompareTo(y.Bar2);
    }
}

To assert that the objects in the two collections are the same and comes in equal numbers (but not necessarily in the same order to begin with), the following lines should do the trick:

var comparer = new FooComparer();
CollectionAssert.AreEqual(
    expectedCollection.OrderBy(foo => foo, comparer), 
    foundCollection.OrderBy(foo => foo, comparer), comparer);    

No, NUnit has no such mechanism as of current state. You'll have to roll your own assertion logic. Either as separate method, or utilizing Has.All.Matches:

Assert.That(found, Has.All.Matches<Foo>(f => IsInExpected(f, expected)));

private bool IsInExpected(Foo item, IEnumerable<Foo> expected)
{
    var matchedItem = expected.FirstOrDefault(f => 
        f.Bar1 == item.Bar1 &&
        f.Bar2 == item.Bar2 &&
        f.Bar3 == item.Bar3
    );

    return matchedItem != null;
}

This of course assumes you know all relevant properties upfront (otherwise, IsInExpected will have to resort to reflection) and that element order is not relevant.

(And your assumption was correct, NUnit's collection asserts use default comparers for types, which in most cases of user defined ones will be object's ReferenceEquals)

Using Has.All.Matches() works very well for comparing a found collection to the expected collection. However, it is not necessary to define the predicate used by Has.All.Matches() as a separate function. For relatively simple comparisons, the predicate can be included as part of the lambda expression like this.

Assert.That(found, Has.All.Matches<Foo>(f => 
    expected.Any(e =>
        f.Bar1 == e.Bar1 &&
        f.Bar2 == e.Bar2 &&
        f.Bar3= = e.Bar3)));

Now, while this assertion will ensure that every entry in the found collection also exists in the expected collection, it does not prove the reverse, namely that every entry in the expected collection is contained in the found collection. So, when it is important to know that found and expected contain are semantically equivalent (i.e., they contain the same semantically equivalent entries), we must add an additional assertion.

The simplest choice is to add the following.

Assert.AreEqual(found.Count() == expected.Count());

For those who prefer a bigger hammer, the following assertion could be used instead.

Assert.That(expected, Has.All.Matches<Foo>(e => 
    found.Any(f =>
        e.Bar1 == f.Bar1 &&
        e.Bar2 == f.Bar2 &&
        e.Bar3= = f.Bar3)));

By using the first assertion above in conjunction with either the second (preferred) or third assertion, we have now proven that the two collections are semantically the same.