If a country had a genuine completely flat income tax system, then it wouldn't matter who paid the tax since it doesn't depend on the employee's other income. Since not many countries run this, it doesn't really make sense for the employee to "take the burden" of the tax, as opposed to merely doing the administration and paying the (probable) amount of tax at payroll, leaving the employee to use their personal tax calculation to correct the payment if necessary.
Your prospective employer is probably saying that your tax calculation in Singapore is so simple they can do it for you. They may or may not need to know a lot of information about you in order to do this calculation, depending what the Singapore tax authorities say.
If you're not a Singapore national, they may or may not be relying on bilateral tax agreements with your country to assert that you won't have to pay any further tax on the income in your own country. It's possible they're merely asserting that you won't owe anything else in Singapore, and in fact you will have taxes to report (even if it's just reporting to your home tax authority that you've already paid the tax). Still, for a foreign worker a guarantee you won't have to deal with the local tax authority is a good thing to have even if that's all it is.
Since there doesn't appear to be any specific allowance for "tax free money" in the Singapore tax system, it looks like what you have here is "just" the employer agreeing to do something that will normally result in the correct tax being paid in your behalf. This isn't uncommon, but it's also not exactly what you asked for. And in particular if you have two jobs in Singapore then they can't both be doing this, since tax is not flat. The example calculation includes varying tax rates for the first X amount of income that (I assume without checking) are per person, not per employment. Joe's answer has the link.
In practice in the UK (for example), there are plenty of UK nationals working in the UK who don't need to do a full tax return and whose tax is collected entirely at source (between PAYE and deductions on bank interest and suchlike). In this sense the employer is required by law to take the responsibility for doing the admin and making the tax payments to HMRC.
Note that a UK employer doesn't need to know your circumstances in detail to make the correct payroll deductions: all they need is a so-called "tax code", which is calculated by HMRC and communicated to the employer, and which basically encodes how much they can pay you at zero rate before the various tax rate tiers kick in. That's all the employer needs to know here for the typical employee: they don't need to know precisely what credits and liabilities resulted in the figure.
However, these employers still don't offer empoyees a net salary (that is, they don't take on the tax burden), because different employees will have different tax codes, which the employer would in effect be cancelling out by offering to pay two people the same net salary regardless of their individual circumstances.
The indications seem to be that the same applies in Singapore: this offer is really a net salary subject to certain assumptions (the main one being that you have no other tax liabilities in Singapore). If you're a Singapore millionaire taking that job for fun, you might find that the employer doesn't/can't take on your non-standard tax liability on this marginal income.