Permit me to try :

Maasi directly refers to the tamil calender month with a period spanning between 15 February & 14 March. Its an auspicious period of Shivites as Shivaratri falls during this time. So Maasi is widely connected or symbolised when talking about Shiva, that you can find it in song usages. But it takes various meaning while combining it with the word that follows

"Maasil veenaiyum malai mathiyumum" is from Thirunaavukarasar Thevaram. Here is means the melody. The same combining with the evening states the wonderful atmospheric weather during the said period. So throughout the depiction of the verse, Maasil stands a call as the descriptive be it time period, adjective or symbolism to God himself.

Other subtle description can be found in a Duets like "Masila Nilave nam" or "Masila unmai Kadhale" : here its refers to Lovely. The moon is lovely during this period or she is lovely as this time of the year. Actually, it has to be read with the context of the narrative, and see the wonderment in the change of its usage.

In Kochadaiiyaan, it lauds Lord Shiva : further details, I invite you to look into Thevaram.

[Anything wrong or found wrong, I stand corrected]