One-on-One vs. One-to-One
Please advise, what is the difference between one on one and one to one?
Is it British vs American English issue?

e.g. When I have a feedback session with my supervisor is it one on one or one to one? Please note, he gives me a performance review and he also wants to hear my feedback about the job. So it is a two way communication. I have always thought this would be a one-on-one. But my company(British) is using the term one-to-one.

Question

One-on-One vs. One-to-One
Please advise, what is the difference between one on one and one to one?
Is it British vs American English issue?

e.g. When I have a feedback session with my supervisor is it one on one or one to one? Please note, he gives me a performance review and he also wants to hear my feedback about the job. So it is a two way communication. I have always thought this would be a one-on-one. But my company(British) is using the term one-to-one.

Answer

These expressions have exactly the same meaning, and you are right in thinking that it is a British/American distinction.

Here in the UK, we tend to say  'one-to-one'.  I remember hearing 'one-on-one' for the first time some years ago, but it has become more common recently.

Answer

I have heard one-on-one more often than one-to-one here in the US (California).

It doesn't refer to the direction of communication, but basically means you are meeting privately with one other person. And it's primarily used in a business setting.

Answer

In practice, there is no difference in meaning.
I think you will find that it is more specific to a company than to a nationality. I have worked for companies based in various countries and I have seen both. 
My last company was Canadian and used "one-on-one". However I view them as synonymous in this context.

Answer

I, personally, use these expressions in different situations.

If two people are having a conversation, then I say they are having a one-on-one discussion.

Or if two people are playing a game of basketball against each other, I say it's a one-on-one game. This can apply to two on two or three on three as well for competitions.

If I am talking about ratios then I say one-to-one. Example:  Private swimming lessons can be a one-to-one teacher-student ratio.