Adding months is where date arithmetic stops being obvious, because months are not a fixed length. The rule used here is the calendar one: the day number is kept and the month advances, so 15 January plus one month is 15 February. Where the day number does not exist in the target month, the result falls back to its last day — 31 January plus one month gives 28 February, or 29 in a leap year.
That fallback is not reversible. Take 31 January, add a month to reach 28 February, subtract a month again and you arrive at 28 January, not where you started. Nothing has gone wrong; there is no arrangement of calendar months in which every day number survives the round trip, which is why some agreements specify 30 days rather than one month.
In the UAE the unit matters because so much is written in months — tenancy terms, service agreements, instalment schedules, probation periods — and month-end start dates are exactly where two systems will produce different answers from the same clause. A period stated in Hijri months is different again: twelve of them come to about 354 days against 365, so a year expressed in one calendar is not the same length as a year expressed in the other.