More generally, how is Singapore's Gini coefficient computed?
According to the Singapore's ministry of finance (MOF):
Singapore’s Gini coefficient is typically reported based on household income from work per household member. The calculation by the Organisation for Economic Co-operation and Development (OECD) is based on the Square Root Scale. ...
... Singapore’s Gini is based on household income from work whereas data on OECD economies is based on income from all sources (which includes non-work income from investments and property).
Their full report on "Income Growth, Distribution and Mobility Trends in Singapore" has more details and you can find even more details in the sources reported therein.
To what extent is it comparable (or not comparable) to those reported by other countries?
There is no simple way of measuring 'comparability'. Generally speaking, any income distribution related cross-country comparisons are problematic and can't be taken at a face value even if the final statistics comes from the same source where the same methodology is followed (like OECD/World Bank) because even if the same methodology is followed when constructing the inequality statistics the raw data that are produced on national level might have some systemic biases as the data come from either household surveys or tax records and different countries might systematically differ in their response rate, rates of tax evasion and other sampling problems/errors (see discussion of this in Atkinson's Inequality book).
This being said despite above difficulties scientists use these statistics in cross-country comparisons (see one such example in Frazer; 2006). So as long as you just don't take these statistics at face value they are still useful for research purposes and also just plain international comparisons. As in the Economist article you linked you can make reasonable guesses about how the differences in methodology or any underlaying data issues affect the statistics and take that into account.