how to get shortname of month in moment.js?

You are not using correct format string, use MMM instead of MMMM see Docs

console.log(moment().format('DD/MMM'))

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As an addendum to Satpal's answer, a simpler (kind of faster/safer approach) would be to go with the Localized formats because preferred formatting differs based on locale.

moment().format('ll') - Sep 4, 1986

Where as

moment().format('LL') - September 4, 1986

Optionally to include time:

moment().format('lll') - Sep 4, 1986 8:30 PM

Where as

moment().format('LLL') - September 4, 1986 8:30 PM