Давам отговор на себе си, защото след дълги лутания, стигнах до този резултат. Решението използва метод DateTimeSpan, който е въведен в класа на праграмата, извън метода Main. (Тук трябва да благодаря на  (VasilValchanov и на https://stackoverflow.com/questions/4638993/difference-in-months-between-two-dates.

Кърк Уол)  

Надавам се да бъде полезно на някой, който търси отговор за "Интервал между дадена дата и днешния ден"!

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

namespace Zaplata
{
    class Program
    {
        static void Main(string[] args)
        {
            Console.Write("Дата днес е: ");
            Console.WriteLine(DateTime.Now);
            Console.Write("Дата на постъпване на работа: ");
         
            DateTime DataEntrance = DateTime.Parse(Console.ReadLine());
            DateTime todaydata = DateTime.Today;
            var dateSpan = DateTimeSpan.CompareDates(DataEntrance, todaydata);
            Console.WriteLine(" Прослуженото време е: "+dateSpan.Years + " Years, " + dateSpan.Months+ " Months, "  + dateSpan.Days+ " Days");
           
            Console.WriteLine();           

        }

        public struct DateTimeSpan
        {
            private readonly int years;
            private readonly int months;
            private readonly int days;
            private readonly int hours;
            private readonly int minutes;
            private readonly int seconds;
            private readonly int milliseconds;

            public DateTimeSpan(int years, int months, int days, int hours, int minutes, int seconds, int milliseconds)
            {
                this.years = years;
                this.months = months;
                this.days = days;
                this.hours = hours;
                this.minutes = minutes;
                this.seconds = seconds;
                this.milliseconds = milliseconds;
            }

            public int Years { get { return years; } }
            public int Months { get { return months; } }
            public int Days { get { return days; } }
            public int Hours { get { return hours; } }
            public int Minutes { get { return minutes; } }
            public int Seconds { get { return seconds; } }
            public int Milliseconds { get { return milliseconds; } }

            enum Phase { Years, Months, Days, Done }

            public static DateTimeSpan CompareDates(DateTime date1, DateTime date2)
            {
                if (date2 < date1)
                {
                    var sub = date1;
                    date1 = date2;
                    date2 = sub;
                }

                DateTime current = date1;
                int years = 0;
                int months = 0;
                int days = 0;

                Phase phase = Phase.Years;
                DateTimeSpan span = new DateTimeSpan();
                int officialDay = current.Day;

                while (phase != Phase.Done)
                {
                    switch (phase)
                    {
                        case Phase.Years:
                            if (current.AddYears(years + 1) > date2)
                            {
                                phase = Phase.Months;
                                current = current.AddYears(years);
                            }
                            else
                            {
                                years++;
                            }
                            break;
                        case Phase.Months:
                            if (current.AddMonths(months + 1) > date2)
                            {
                                phase = Phase.Days;
                                current = current.AddMonths(months);
                                if (current.Day < officialDay && officialDay <= DateTime.DaysInMonth(current.Year, current.Month))
                                    current = current.AddDays(officialDay - current.Day);
                            }
                            else
                            {
                                months++;
                            }
                            break;
                        case Phase.Days:
                            if (current.AddDays(days + 1) > date2)
                            {
                                current = current.AddDays(days);
                                var timespan = date2 - current;
                                span = new DateTimeSpan(years, months, days, timespan.Hours, timespan.Minutes, timespan.Seconds, timespan.Milliseconds);
                                phase = Phase.Done;
                            }
                            else
                            {
                                days++;
                            }
                            break;
                    }
                }

                return span;
            }
        }
    }
    
}