1. No, it isn't a plural. It's one person or animal. In fact. 'their' isn't always plural - it can be used to refer to a singular subject. Nowadays, 'their' is the accepted possessive adjective to refer to 'someone', 'anybody', 'nobody', 'a person' 'whoever', or any other singular subject whose gender is unknown. We don't know whether this 'someone' is male or female, so we use 'their'.
2. Сan I use the words: "own", "his" or "its" instead "their" ?
Can you use 'own'? No. The word 'own' is not a possessive form. You can add 'own' if you want - Someone is proud of their own work - but it can't replace 'their'.
Can you use 'his'? Only if you know that the subject is male.
Can you use 'its'? No, because 'its' doesn't fit with 'someone'.
'Their' is the only word that sounds right here. It's quite common to use 'they', 'them' and 'their' with a singular meaning, and this is actually a very useful way of phrasing things when you don't know, or it doesn't matter, whether the subject is male or female. For example . "I don't know who the last person was who used this shower, but they left their soap and towel there'.
If you scroll down the articles on this site, you'll find one on this subject.