why it has the same meaning between "Oes draig gyda chi?" and "Oes gynnoch chi ddraig?"
Because Welsh people decided so?
Some Welsh speakers use gyda, which does not change and which goes after the noun, while others use gan, which changes form according to the subject and which goes before the noun.
It's perhaps a bit like how some English speakers ask "Do you have a dog?" and others ask "Have you got a dog?". They mean the same thing, even though the words used are a bit different and the order is a bit different -- some speakers use one form, some use the other one.
Double weirdness here: I tried "Do you own a dragon" as an answer, which aparently was wrong. The suggested "correct" answer was "Do you've a dragon?" That makes no sense in English.
That is a fault in the general Duo software. You should report it in general 'Troubleshooting' forums.
'Owns' in Welsh uses a different, emphatic pattern with the irregular verb piau/biau. That is covered elsewhere in the course.
We have three possible correct answers entered here:-
Do you have a dragon?
Have you got a dragon?
Have you a dragon?
The system should have given you the nearest correct answer to your suggestion, ie the one above. As ibisc suggests there seem to be general glitches appearing in the system at the moment after the recent major overhaul of the system.