He's one of, if not the furthest left member of his cabinet. When he formed the government the only way he could get to 61 in the Knesset (required for a majority) was to hook up with the right wing whackadoos nobody else would touch. I know it sounds nuts in today's world but he's the sanest of the bunch as fas as I'm concerned.
There is no "vice-prime minister". If he goes, technically the government falls. The existing cabinet appoints a new prime minister from their midst who can serve up to 100 days. New elections are called and the top party is invited to to form a new government. If that doesn't happen in 60 days (I believe), the #2 guy gets a shot at it. If that doesn't work, it's back to the polls and it starts all over again. The current government was formed after 5 elections spanning a 3+ year period from 2019 to 2022. And through that whole period, Bibi (the incumbent) was prime minister (either officially or in effect).
If Bibi bites it, the whackadoos pick the new prime minister who sits for 100 days or until a new government if formed. While it's unclear, it appears that if no government can be formed, on day 101 the whadadoos pick another prime minister who sits for another 100 days or until a new government is formed. And so on and so on...
To put it in perspective, translate that to the US. If we had a similar succession approach, Hegseth, Bondi, Vance, Rubio, RFK jr, and the lot would pick among themselves who would be in charge for the next 100 days. Wouldn't that be fun.
PS - we have a similar experience in the US. Elliot Richardson delayed going after Nixon knowing that Agnew would be his replacement. He knew that before he could press any case against Nixon, he had to purge Agnew who, in his mind, was far worse than Nixon. It's not fully analogous but it rhymes.