There are two ways of using Do-Re-Mi:
1) the 'fixed-do' system, which is an absolute reference to the notes on the keyboard (Do = C).
2) the 'movable-do' system (as mentioned by Pianotechenthusiast), used by singing teachers internationally, even in countries which use alphabetical note names. 'Do' is set as the key note of the key you're singing in, for example F (so in this case Do Re Mi would be F G A). I imagine this can get incredibly confusing for students who learn the notes as Do Re Mi, especially if they have perfect pitch.

In the Rogers & Hammerstein song, I'm pretty sure they have in mind the movable-do system. 'Ti' instead of 'Si' tends to be preferred by English-speaking musicians - its origin seems to have been from a teacher who wanted to represent each step with a single letter: D R M F S L T D.
The 'meanings' given to the solfege names were just made up by Hammerstein to make memorable rhymes! Even so, it's a very good song for learning solfege - a modern equivalent of Ut Queant Laxis. Each phrase begins on the next step of the scale.

  How they translate that for each idiom? In my country, Brazil, we use Do-Re-Mi-Fa-Sol-La-Si (in the movie wad... Sol-La-Ti). We have a translated version to portuguese of J. Andrews lyrics. Of course the meaning given for each note is another as they had to find something to fit.

https://youtu.be/drnBMAEA3AM

I used do re mi fa so la ti (not si) so that so could be G and si G#.  C do, C# di, D re, D# ri, E mi, F fa, F#  fi, G so, G# si, A la,  A# li, B (H) ti, C do.  solfa chromatic
Stig

We use this system in the Netherlands, "i" is for sharp, "e" is for flat.
G# we sing "si".
Gb we sing "se"
except Db, we sing "ra"

best wishes
k.c.Paul

For example, in the key of “C Major”, “Do” is “C”, but in the key of “F Major”, “Do” is “F”. Moveable.

With this system they could learn new simple melodies they never heard before, and using solfege hand signs. https://www.musictheorytutor.org/2013/0...and-signs/

Practicing this required music lessons every day so at that time I was privileged to have 1 hour of music every day (weekly schedule reduced maths, languages). After two years they could sing 2 and 3 voice songs.
But this was 45 years ago   

All the best, everyone,

Stig

Thinking about different systems of naming musical notes (CDEFG... or Do Re Mi... etc.) led me to this fascinating resource: https://www.reddit.com/r/lingling40hrs/...the_world/
Then I remembered that, in French, the key of C major is expressed as 'Ut majeur'. After a little bit of digging, it seems that 'ut' is an academic convention, used (for example) to signify the key of C rather than the note C (Do). Is that correct?

Yep, all this comes from the middel ages, as a latin hymn (Gregorian chant) where it was supposedly used to teach music in the monasteries, because the verses used  successive notes of the scale. Here it's very well explained:

Wikipedia Link

BEST REGARDS