View

<Window x:Class="WpfApp1.MainWindow"
        xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
        xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
        xmlns:d="http://schemas.microsoft.com/expression/blend/2008"
        xmlns:mc="http://schemas.openxmlformats.org/markup-compatibility/2006"
        xmlns:local="clr-namespace:WpfApp1"
        mc:Ignorable="d"
        Title="MainWindow" Height="450" Width="800">
    <Window.DataContext>
        <local:ViewModel/>
    </Window.DataContext>
    <Grid>
        <ListView HorizontalAlignment="Left" Height="211" Margin="162,94,0,0" VerticalAlignment="Top" Width="414" ItemsSource="{Binding Path=Persons, UpdateSourceTrigger=PropertyChanged}" SelectedItem="{Binding SelectedItem, Mode=TwoWay, UpdateSourceTrigger=PropertyChanged}">
            <ListView.View>
                <GridView>
                    <GridViewColumn Header="Name" DisplayMemberBinding="{Binding Name}"/>
                    <GridViewColumn Header="Age" DisplayMemberBinding="{Binding Age}"/>
                </GridView>
            </ListView.View>
        </ListView>
        <Button Content="Dodaj" Command="{Binding AddCommand}" HorizontalAlignment="Left" Margin="162,52,0,0" VerticalAlignment="Top" Width="75" Height="37"/>
        <Button Content="Wyczyść" HorizontalAlignment="Left" Command="{Binding ClearCommand}" Margin="242,52,0,0" VerticalAlignment="Top" Width="75" Height="37"/>
        <Label Content="{Binding SelectedItem.Name}" HorizontalAlignment="Left" Margin="162,310,0,0" VerticalAlignment="Top" Width="414"/>
    </Grid>
</Window>

ViewModel

using System;
using System.Collections.Generic;
using System.Collections.ObjectModel;
using System.Collections.Specialized;
using System.ComponentModel;
using System.Linq;
using System.Runtime.CompilerServices;
using System.Text;
using System.Threading.Tasks;
using System.Windows.Controls;
using System.Windows.Input;

namespace WpfApp1
{
    public class Person
    {
        public string Name { get; set; }
        public int Age { get; set; }
    }

    public class ViewModel : INotifyPropertyChanged
    {
        public ObservableCollection<Person> Persons { get; } = new ObservableCollection<Person>();

        private Person _previousPerson;
        private Person _selectedItem;

        public Person SelectedItem
        {
            get { return _selectedItem; }
            set
            {
                if (value != null)
                {
                    _previousPerson = value;
                }
                Set(ref _selectedItem, value);
            }
        }

        public ICommand AddCommand { get; private set; }
        public ICommand ClearCommand { get; private set; }


        public ViewModel()
        {
            AddCommand = new Command(() =>
            {
                var names = new[]
                {
                    "Damian",
                    "Adam",
                    "Michał",
                    "Piotrek",
                    "Agnieszka",
                };
                var random = new Random();
                var person = new Person
                {
                    Name = names[random.Next(names.Length)],
                    Age = random.Next(18, 60),
                };

                Persons.Add(person);
            }, () => true);

            ClearCommand = new Command(() => Persons.Clear(), () => true);

            Persons.CollectionChanged += (sender, args) =>
            {
                if (args.Action == NotifyCollectionChangedAction.Add)
                {
                    if (_previousPerson != null)
                    {
                        var item = Persons.Where(x => x.Name == _previousPerson.Name).FirstOrDefault();
                        if (item != null)
                        {
                            SelectedItem = item;
                        }
                    }
                }
            };
        }

        protected virtual bool Set<T>(ref T instance, T value, [CallerMemberName] string propertyName = null)
        {
            if (EqualityComparer<T>.Default.Equals(instance, value))
                return false;

            instance = value;
            OnPropertyChanged(propertyName);
            return true;
        }

        public event PropertyChangedEventHandler PropertyChanged;

        protected virtual void OnPropertyChanged([CallerMemberName] string propertyName = null)
        {
            PropertyChanged?.Invoke(this, new PropertyChangedEventArgs(propertyName));
        }
    }
}