Twój seems to be suggested answer, but maybe you got alternate "wasz" instead.
Poor English language does not have different words for ty=1 person and wy=more than one person.
You had thou for ty, and sometimes use you all for wy.
twój is your = belonging to "ty", one person
wasz is your = belonging to "wy", 2+ people
no. you use pan/pani for formal speech. using wy as a polite form exists only in some dialects but not in standard Polish
Hi, can I also say Swój tygrys? And, if yes, which is most natural in spoken Polish?
No, "swój" is almost non-existent in Nominative, and I only say 'almost' because of some rare colloquial usage. The only thing that comes to my mind is "swój chłop", which is rather hard to translate... that's someone that "is okay", "can be trusted", something in that direction.
Apart from this digression, which I only mention because I know your Polish is already quite good, no, you can't say that. Why? Because "swój" refers to the subject of the sentence. And there is no other subject to refer to. "swój tygrys" would be like "tiger's tiger"... although perhaps it makes even less sense.