You're right that the opposite of hate is not love, but apathy. In my experience, people hate someone for two main reasons:

1) Rejection - "I hate that girl/boy because she/he dumped me."

2) The person they "hate" reflects things about themselves that they don't like, but don't admit.

I'm 100% sure that a huge proportion of the "hate" of Putin is like the "hate" for Trump. Trump does things and says things that people who hate him also do, but are not proud of.

There is nothing that "Putin" (and by "Putin" I mean Russia because the West and particularly America tries to reduce the largest country on the planet to a single individual) has done in Ukraine that the US has not done 5 or 6 times in the last 30 years.

So many Americans "hate" Putin because our propaganda has helped us things like if we starve 500,000 children in Iraq then "it was worth it" (in the immortal words of Madelaine Albright) - and note this is before the US unprovoked invasion.

That was obviously a horribly barbaric act and IMO a war crime. When Russia is accused of something similar it is also horrible, but rational humans can see our hypocrisy and we hate Russia and Putin for making our hypocrisy so clear.