Is it common to say “ have higher preference for something.”? No, you would not use this on spoken English. It is too formal. Use it in written... 'Prefer' is much more natural...

Just preference will do. eg. I prefer to take the private transport as it is more flexible. Do you have any preference of what to eat for dinner later? Cannot use interchangeably with prefer. Different sentence structures use different words.

No need for‘higher’ in the sentence

'higher' is wrong. You can just say 'I have a preference for private transportation ...' 'have a preference for' = 'prefer'