How ownership is registered
When you own listed shares, these are usually registered automatically with your bank or online broker, managed centrally via a CSD like Euroclear.
For unlisted shares, the company's own board of directors is responsible for maintaining a share register (aktiebok). You must be listed here to be considered the formal owner.
Share register vs Share certificates
A share certificate is just physical (or digital) proof of ownership, but the share register is always the legally binding document. If you buy shares, the company must enter you into the share register before you have the right to vote at general meetings.
Summary
Always ensure that you or your holding company have been entered into the company's share register following a transaction.