Postdoc interview - somewhat positive reply but no news?

One possible scenario is that they have a preferred candidate, who has not yet accepted or rejected their offer.

Another possible scenario is that there is an uncertainty about funding. For example, they may be trying to hire you using some complicated scheme in which your position is co-funded by a third body. It takes time to organize such things. Or they may be waiting for an announcement of the outcome of their grant application submitted long ago. They may have decided to hire you anyway, but may not yet know which grant to fund your position from. Or they may have been unable to hire you in the intended way and may be looking for an alternative method to hire you.

It is rather a bad sign that they did not take the initiative to contact you and keep you updated, and such a delay seems to be too long to be caused by any official process unless this process is related to an application for some funding.

If I were you, I would be rather pessimistic and would send a frank email expressing my concern. I would ask them to frankly explain what is going on and to tell me the probability of me getting an official offer. I would also ask them when I should expect to receive an official offer. If I did not receive a convincing and credible response, I would immediately start looking for another position.

The PI told you that you got the job, so you should contact them again. I'd suggest you start being more pressing, something along the lines of "I would like to know when the contract will start, since I have to make arrangements..."

It's quite common for the internal hiring process to take a long time so there's no reason to be alarmed, but you've been quite patient already. In case you have other opportunities available you can mention it, it might speed things up if they worry that they might lose you.