Basically I would get the job if I asked for it, but I don't know the salary yet. That is what it is. Then I would get a formal offer with salary, benefits, etc.
Job offer with no terms, salary, starting date is not a job offer; it's nothing, and that's all you have now. It's important to get that clear and out of the way that we are at the same level - unless it's a job offer, in writing, with all the details and already signed by the other party you got absolutely nothing.
Thus...
Should he "accept" that job offer and try to negotiate the salary based on what his current salary is?
There is no job offer to accept... Yet. So lets get to the crux of what the question is all about:
How to handle a situation where I, maybe, can get a better job somewhere else, but I am currently employed and want to be very careful, as nothing is set in stone yet.
I wouldn't handle this as any other interview/job ad. Reach out, ask all the questions you want to know, like salary, and when you have all the information and it seems worth the effort ask to apply and go through whatever process there is.
If the process truly is as simple as "I just have to ask", great! You are not far off. But if that's the case, no one should be surprised by a question of nature "hey, if I were to join you as ACME maintenance officer, what would my pay be?".
And if that matches your interest, proceed with the application (as informal as it may be) and hopefully at the end of it there's an actual job.