How can I tell if a person or animal is actually a jinn?

There is no such thing as a test to figure out whether something is a jinn. There is nothing of the sort in Quran or Hadith or anywhere else.

Proof?

In the conditions of Marriage, there is no requirement that you must test your bride to see if she is human.

In the conditions of Halal Zibah and Hunting, there is no requirement that you must test the animal to see if what you are about to kill and eat is anything other than what it appears to be.

If such a test existed, it would have been prescribed in the conditions of the above.

Jinn can't "become physical beings" per se, and by "physical" I presume is meant creatures that humans can perceive and have a tangible existence. Rather they inhabit or "possess" physical beings, which in this case would not mean "own" but "take over". Thus it is that they can also be cast out or exorcised by an Imam or Sheikh trained in doing this and who should have strong faith and a strong personality to withstand the difficulties involved. There are specific protocols, including the reading of specific ayat from the Quran a specific number of times, as well as other things to be said.

As to how one can "identify" a case of possession, usually the victim, who should have been previously normal and without mental or behavioral issues, suddenly or gradually takes on abnormal behaviors which would always be negative and destructive. They would tend to steer away from Islam and salat if they were Muslim, to treat family and friends with hostility, isolate themselves, and also engage in self-destructive behavior. Physical manifestations could include speaking in an abnormal voice or voices, acting out with violence, using profanity, and sleep disturbances such as nightmares. It is the combination of all of this into a noticeable and frightening trend that clearly identifies the victim as a potential case of possession.

As for animals, a possessed animal would act violently against people, but this is not a definitive matter because animals can act that way for many other reasons. Jinn are interested in harming people, not animals, so the point of taking over an animal would be to terrify and harass people.

So it is not true even theoretically that "any animal or person can be a jinn." The Quran mentions some people as being Satanic, but this is due to their bad behavior reaching a point of no return, not jinn possession. They are referred to as "shaiateen al-ins" or "satans in human form." They are humans who have so committed themselves to evil that they seek to destroy others' lives as well and work on Satan's revenge as an avowed enemy of humankind. Of course, such people are their own worst enemy.