This achievement is slightly misleading. It does say 5 days in a row, HOWEVER, they can't be 5 days in a row from the calendar.
For example, if it's the 10th day of the month, and you go into the calendar and do a challenge on the 1st, 2nd, 3rd, 4th, and 5th of the month, the achievement won't pop.
What you need to do, is do the daily challenge five days running, so, for example, if it's the 10th day of the month, you want to do one of the daily challenges for the 10th on the 10th. Then, the following day, you want to do one of the daily challenges of the 11th, on the 11th. Then one of the daily challenges of the 12th, on the 12th. Basically, so on and so forth until you've done it five days in a row. Or in simple terms, using the 10th as Day 1:
Day 1 (10th of month): Do one of the 3 challenges for the 10th.
Day 2 (11th of month): Do one of the 3 challenges for the 11th.
Day 3 (12th of month): Do one of the 3 challenges for the 12th.
Day 4 (13th of month): Do one of the 3 challenges for the 13th.
Day 5 (14th of month): Do one of the 3 challenges for the 14th.
When you've done a challenge on the 5th day running, the achievement will pop when your challenge coins are rewarded.
Please note, that you do NOT need to do ALL THREE challenges on each day for the 5 days that this achievement involved, you only need to do a minimum of one per day.
I do however, recommend you try and do all of the challenges, as the coins you get contribute to the bronze/silver/gold medal longevity achievements.
Also, please note, that you should not do challenges of the day from the calendar. To make sure they count, before doing any of the day's challenges, restart the game, click on "Daily Challenges", and do them from the page that comes up. Do NOT go to the calendar to do them.