For everything where identity is important there is liberal use of an ID card of decent quality, one with several counterfeiting features. Any drivers license also doubles as one.
Online an electronic signature of some kind is used, exactly which varies a bit, but the one called BankID is accepted in most places.

Long rambling post ahead, I'm really, really terrible at writing something mostly coherent on the phone, apologies to everyone, but the above paragraph really is the only important part of this comment :)

For bigger financial transactions, if your counterpart is a company they will often do a credit check, which the credit checking provider is then by law required to inform you about through regular mail.

Sensitive online services from government and other companies mostly use some version of an electronic ID, the most common is generally known as BankID, despite of its name it's simply an embodiment of an electronic ID. BankID got its name because it is provided by Bankgirot, a clearinghouse company with a long history in which most of the major Swedish banks have some ownership. Another implementation is provided by Telia AB, a formerly state owned telecommunications provider.

It's not a bulletproof system by any means, but for the most part it appears to work.

However, I'm not too enthusiastic about the eagerness to adopt "BankID on your mobile phone”. It feels like it's only a matter of when, rather than if, someone will manage to exploit it through som security flaw in some of the popular smart phones. Which will inconvenience a lot of people until banks - most likely - simpy roll back the transactions, as is in the digital realm it's hard to disappear money when literally everyone gets assigned an ID.

Tangentially, the most common way used to create legitimate companies for illegal activities AFAIK, at least used to be to pay of some substance abuser or petty thief to use their identity, and simply have them take the fall when/if things go south. So apparently, creating fake identities would appear to be rather hard.

Because the ubiquity of the personal number, and the liberal use of good quality ID cards. This number connected through tax, loans, land ownership, insurance etc, makes it somewhat hard to make money leave the digital world without trace. It would also be terrible in the hands of a competent totalitarian regime or fundamentalist (of any sort) government. Sometimes I think we subconsciously avoid that by voting for mostly uncharismatic leaders with little appeal except for them promising a little bit more money in your wallet, better social security, healthcare, child care, or school. In all cases better can be read as more available, efficient, and ecological from the left - and higher fees, stricter requirements, and lower availability from the right. But I digress.

There have been some areas of law/contract where identity theft was easier, notably land ownership/property laws. IIRC in the property laws it was essentially stated that a signature on paper with two witnesses signatures was indirectly used as proof of identity. Unsurprisingly, this led to some people getting their homes sold by third parties, and having to face a rather arduous legal procedure to get back to square one.

Yeah, that one was rather monumentally stupid, but it wasn't really caused by the personnummer being public.