Salam alaykum wa rahmatullah.

Muharram is mourning month for shias. Because sayidina Hussayn (r.a) was killed in muharram.
I know that shias believe all 11 aimma were murdered.

Ali was killed in 21 Ramadan.
Hasan ibn Ali in 7 Safar
Zeynal Abeeden in 25 Muharram
Al Baqir in 7 Dhul hijja
Sadiq in 25 Shawwal.
Al Kadheem in 25 Rajab
Ar Rida in 30 Safar.
Jawad in 30 Dhul-Qada
Hadi in 3 Rajab.
Askari in 8 Rabi awwal. (may Allah be pleased with all of them) Dates taken from book of Shirazi “The Shi’a and their beliefs” p 50-51.

Let me summarize.

2 aimma died in muharram.
2 in rajab.
2 in safar
1 in ramadan.
1 in dhul hijja.
1 in shawal.
1 in dhul qada.
1 in rabiul awwal.

U see, what is the question? Why only muharram is mourning month for shias? Or may be i don’t know, and all these 8 moths are mourning one?