There are names of months in Islam because the Islamic calendar started with the event of Hijrah. There was also a need to mark the special months:
They ask you concerning the new moons (Literally: crescents). Say,
"They are fixed times for mankind, and (for) the Pilgrimage." ... (Surah Al-Baqarah, 189)
Because existing methods were being misused:
Indeed, the postponing
[of restriction within sacred months] is an increase in disbelief by
which those who have disbelieved are led [further] astray. They make
it lawful one year and unlawful another year to correspond to the
number made unlawful by Allah and [thus] make lawful what Allah has
made unlawful. Made pleasing to them is the evil of their deeds; and
Allah does not guide the disbelieving people. (Surah Al-Tawbah,37)
As for days, they already had their names in Arabic version. Only one of the days of week is special and is marked i.e., Al-Jummah(Friday).
You could refer Islamic new year beginning in Muharram to know more about the starting of Islamic calendar.