The reason why I'm thinking Rinoa talking to Squall is this:

Hello Stranger, I don't know you.
Care to try your luck?
First meeting at the 'graduation ball'. They don't know each other.

Try to keep up with me,
you'll be in for a shock.
Dancing lessons. Rinoa guiding Squall though a dance, a few times.

Goodbye Stranger, I don't know you,
it was fun for a bit,
Rinoa finds the dance fun, yet she has to depart for a reason.

you're actually quite dazzling,
I really must admit.
Rinoa found Squall's dancing - once he got the hang of it - quite dazzling.

Hello Stranger, I don't know you,
here's what we will do.
Rinoa on the Forest Owl's train in Timber. Explaining the 'switcharoo'.

You go down and open the lock,
we'll pull the switcharoo.
The switch of the presidents.

Finally Stranger, now I know you.
You were such a sight.
Not to sure what this means, other then Rinoa finding Squall handsome.

I am stricken, and if you ask,
I'll follow you into the night.
Rinoa is stricken with the "Sorceress Powers", which means that she would 'follow' Squall 'into the night'. Or Rinoa could be stricken with love for Squall.

Who am I?
Rinoa