Short answer: you don’t use CallContext since it’s not available to either .NETStandard libraries or .NETCore apps ;-).

Long answer: after landing at the same question on StackOverflow, learning about the only potential answer, it’s pretty straight-forward actually.

Just create the following static class mimicking the CallContext API:

    /// <summary>
    /// Provides a way to set contextual data that flows with the call and 
    /// async context of a test or invocation.
    /// </summary>
    public static class CallContext
    {
        static ConcurrentDictionary<string, AsyncLocal<object>> state = new ConcurrentDictionary<string, AsyncLocal<object>>();

        /// <summary>
        /// Stores a given object and associates it with the specified name.
        /// </summary>
        /// <param name="name">The name with which to associate the new item in the call context.</param>
        /// <param name="data">The object to store in the call context.</param>
        public static void SetData(string name, object data) =>
            state.GetOrAdd(name, _ => new AsyncLocal<object>()).Value = data;

        /// <summary>
        /// Retrieves an object with the specified name from the <see cref="CallContext"/>.
        /// </summary>
        /// <param name="name">The name of the item in the call context.</param>
        /// <returns>The object in the call context associated with the specified name, or <see langword="null"/> if not found.</returns>
        public static object GetData(string name) =>
            state.TryGetValue(name, out AsyncLocal<object> data) ? data.Value : null;
    }

Then just make sure it behaves as you expect, by spinning a bunch of tasks, mixed in with good old Threads just for the sake of it, and see the test pass:

        [Fact]
        public void WhenFlowingData_ThenCanUseContext()
        {
            var d1 = new object();
            var t1 = default(object);
            var t10 = default(object);
            var t11 = default(object);
            var t12 = default(object);
            var t13 = default(object);
            var d2 = new object();
            var t2 = default(object);
            var t20 = default(object);
            var t21 = default(object);
            var t22 = default(object);
            var t23 = default(object);

            Task.WaitAll(
                Task.Run(() =>
                {
                    CallContext.SetData("d1", d1);
                    new Thread(() => t10 = CallContext.GetData("d1")).Start();
                    Task.WaitAll(
                        Task.Run(() => t1 = CallContext.GetData("d1"))
                            .ContinueWith(t => Task.Run(() => t11 = CallContext.GetData("d1"))),
                        Task.Run(() => t12 = CallContext.GetData("d1")),
                        Task.Run(() => t13 = CallContext.GetData("d1"))
                    );
                }),
                Task.Run(() =>
                {
                    CallContext.SetData("d2", d2);
                    new Thread(() => t20 = CallContext.GetData("d2")).Start();
                    Task.WaitAll(
                        Task.Run(() => t2 = CallContext.GetData("d2"))
                            .ContinueWith(t => Task.Run(() => t21 = CallContext.GetData("d2"))),
                        Task.Run(() => t22 = CallContext.GetData("d2")),
                        Task.Run(() => t23 = CallContext.GetData("d2"))
                    );
                })
            );

            Assert.Same(d1, t1);
            Assert.Same(d1, t10);
            Assert.Same(d1, t11);
            Assert.Same(d1, t12);
            Assert.Same(d1, t13);

            Assert.Same(d2, t2);
            Assert.Same(d2, t20);
            Assert.Same(d2, t21);
            Assert.Same(d2, t22);
            Assert.Same(d2, t23);

            Assert.Null(CallContext.GetData("d1"));
            Assert.Null(CallContext.GetData("d2"));
        }

You could also have a typed version of CallContext:

    public static class CallContext<T>
    {
        static ConcurrentDictionary<string, AsyncLocal<T>> state = new ConcurrentDictionary<string, AsyncLocal<T>>();

        /// <summary>
        /// Stores a given object and associates it with the specified name.
        /// </summary>
        /// <param name="name">The name with which to associate the new item in the call context.</param>
        /// <param name="data">The object to store in the call context.</param>
        public static void SetData(string name, T data) => 
            state.GetOrAdd(name, _ => new AsyncLocal<T>()).Value = data;

        /// <summary>
        /// Retrieves an object with the specified name from the <see cref="CallContext"/>.
        /// </summary>
        /// <typeparam name="T">The type of the data being retrieved. Must match the type used when the <paramref name="name"/> was set via <see cref="SetData{T}(string, T)"/>.</typeparam>
        /// <param name="name">The name of the item in the call context.</param>
        /// <returns>The object in the call context associated with the specified name, or a default value for <typeparamref name="T"/> if none is found.</returns>
        public static T GetData(string name) =>
            state.TryGetValue(name, out AsyncLocal<T> data) ? data.Value : default(T);
    }

which you would call like:

CallContext<Foo>.SetData("foo", foo);
...
var foo = CallContext<Foo>.GetData("foo");

And it Just Works :D