My question is - should I share that bonus with him ?
No. Not at all. Referral bonuses are common in a lot of industries because they're advantageous for all parties involved. The company finds interesting candidates and saves on the recruiter fees. Candidates are put in touch with good employers and have someone on the inside to get a better sense of the culture and their fit1 And you get a nice bonus for your trouble and for using your network.
There is absolutely zero expectation that you share this referral bonus! Referral bonuses are a standard business practice and it is indeed an unwritten rule that these bonuses are not expected to be shared. That is not to say that there aren't people who share, but that's usually a sign of a much closer friendship than is normal for referrals who are usually more like acquaintances. The problem with sharing is that you're talking about a very significant amount of money which is way, way above the threshold for appropriate gifts between (former) colleagues.
Compare it to winning a modest lottery. If you wouldn't share that with this person, there's no reason to share your bonus. I recently contacted a former colleague when I wanted to apply with his company just so he could get the referral bonus. All he did was take a few phone calls so I could ask some questions from an insider but he's still earned that bonus and I would feel very uncomfortable if he offered to share it. That's just not really done.
As Alison Green said, taking this person out for a meal would be a great idea:
No one here needs to treat anyone. You did your employer a favor by helping to connect them with a good employee; it wasn’t an act of charity toward your friend, so she doesn’t need to treat you. And your employer gave you a finder’s fee because they want good referrals; you’re not obligated to share the cash. That said, if you’d like to take your friend out to celebrate — totally separate from the referral bonus — by all means do!
1 - Not to be confused with bypassing the hiring process which is unethical!