You can make your entities inherit from a common interface:

public interface IEntity
{
    int ID { get; set; }
}

Then you can redefine your method like

public void AddItem<TEntity>(TEntity entity) where TEntity : IEntity
{
    // Now you can access entity.ID
}

Now, if you don't always want to compare via ID, then you can add a predicate to your method:

public void AddItem<TEntity>(TEntity entity, Func<TEntity, bool> predicate)
{
    var objects = GetAllItems().ToList();

    // You might need some logic in the predicate to check for null
    if(!objects.Any(a => predicate(a as TEntity))
    {
        objects.Add(entity);
        _cache.AddReplaceCache(objects);
    }

}

Then you would use your function as

repository.AddItem(entity, e => e.ID == entity.ID && e.OtherProperty == entity.OtherProperty);

如果我没理解错的话，你的问题是TEntity没有属性ID。所以让你的实体继承公共接口，比如ID列。

public interface IObject
{
    int ID {get; set;}

    //define all other properties which are shared between your Entities.
}
public class MyObject : IObject
{
    public int ID {get; set;}

    //other properties.
}

public void AddItem(TEntity item): where TEntity:IObject
{
    var entities = GetAllItems().ToList(); //Method gets cached IEnumerable<TEntity>

    if(!objects.Any(a=> a.ID == item.ID ))//Generically see if TEntity is already in the list based of defined properties
    {
        entities.Add(item);
    }

}