No, long-mustache is not an Islamic custom. Instead, the Islamic custom is to trim the mustache which is established by the Prophet (pbuh) himself.
It was narrated that Anas bin Malik said:
وُقِّتَ لَنَا فِي قَصِّ الشَّارِبِ وَحَلْقِ الْعَانَةِ وَنَتْفِ
الإِبْطِ وَتَقْلِيمِ الأَظْفَارِ أَنْ لاَ نَتْرُكَ أَكْثَرَ مِنْ
"We were given a time limit with regard to trimming the mustache,
shaving the pubic hairs, plucking the armpit hairs and clipping the
nails. We were not to leave that for more than forty days." Muslim
The words for 'trimming the mustache' in the hadith are
قَصّ الشَّارِبِ which literally means
clipping of the drinkers. According to Lisaan-ul-Arab (لسان العرب), the famous Arabic dictionary
ما سالَ على الفَم من الشَّعر i.e. those hair which flow over the mouth. The mustache hair when long tend to touch ("drink") the water we are drinking. That's why they are named so. From this the maximum length can be inferred i.e. the hair should be short enough not to touch the water while drinking.