TimeZoneInfo.ConvertTimeFromUtc is not converting old dates to then daylight saving time

You're using the wrong timezone. "E. Australia Standard Time" corresponds to Brisbane time, and that timezone does not observe DST so your winter and summer times come back with the same UTC offset.

For Sydney, you need to use "AUS Eastern Standard Time". ConvertTimeFromUtc will apply the UTC offset as appropriate for that timezone and give you the times you're expecting.

Here is your snippet including the use of the appropriate time zone, and a bit of editing around the Timezone initialisation so it compiled:

    public void ConvertTimes()
    {
        string Timezone = "AUS Eastern Standard Time";

        TimeZoneInfo ESTZone = TimeZoneInfo.FindSystemTimeZoneById(Timezone);
        DateTime date = Convert.ToDateTime("2019-01-23 13:15:23.6090752");
        DateTime SydneyDateTime = TimeZoneInfo.ConvertTimeFromUtc(date, TimeZoneInfo.FindSystemTimeZoneById(Timezone));
        string EST = SydneyDateTime.ToString() + " / " + SydneyDateTime.IsDaylightSavingTime().ToString() + " / " + ESTZone.IsDaylightSavingTime(SydneyDateTime) + " / " + ESTZone.DaylightName + " / " + ESTZone.StandardName;
        DateTime IndiaDateTime = TimeZoneInfo.ConvertTimeFromUtc(date, TimeZoneInfo.FindSystemTimeZoneById("India Standard Time"));


        DateTime date2 = Convert.ToDateTime("2019-05-23 13:15:23.6090752");
        DateTime SydneyDateTime2 = TimeZoneInfo.ConvertTimeFromUtc(date2, TimeZoneInfo.FindSystemTimeZoneById(Timezone));
        string EST2 = SydneyDateTime2.ToString() + " / " + SydneyDateTime2.IsDaylightSavingTime().ToString() + " / " + ESTZone.IsDaylightSavingTime(SydneyDateTime2) + " / " + ESTZone.DaylightName + " / " + ESTZone.StandardName;
    }

This gives the output:

EST = "24/01/2019 00:15:23 / False / True / AUS Eastern Daylight Time / AUS Eastern Standard Time"

EST2 = "23/05/2019 23:15:23 / True / False / AUS Eastern Daylight Time / AUS Eastern Standard Time"

which I think is what you were expecting.

Incidentally, DateTime.IsDaylightSavingTime() uses your local timezone to determine whether the time is within DST. It might be misleading looking at it here.