It's perfectly normal and common to say that a person "is 20," or "is 47," or any other age, and it's implicitly understood that we're talking about age. This works with animals, too.
But you can't say "That desk is 50," or "My house is 25." You have to specify "That desk is 50 years old" and "My house is 25 years old."
When did we start dropping the "years old" and why did the change only apply to living things?
My answer focuses on the OP's first question: "When did we start dropping the 'years old' [from expressions of people's age]"?
To see when this started happening in published writing, I ran Google Books searches for five expressions—"he is twelve [or 12]," "he is eighteen [or 18]," "he is twenty-five [or 25]," "he is thirty-two [or 32]," and "he is forty-five [or 45]." Here are the earliest matches for each phrase in which the phrase is understood to refer to an age and the phrase is not completed by "years of age" or "years old":
"he is twelve [or 12]": 1619; 1808; 1867
"he is eighteen [or 18]": 1770; 1773; 1803
"he is twenty-five [or 25]": 1837; 1890; 1900
The earliest of these instances is from Beaumont & Fletcher's The Maid's Tragedy (1619):
Amintor. Who's there? my brother! I'm no readier yet./Your sister is but now up.
Diph. You look as you had lost your eyes tonight:/I think you have not slept.
Amintor. I'faith I have not.
Dip. You have done better, then.
Amintor. We ventured for a boy: When he is twelve,/He shall command against the foes of Rhodes./Shall we be merry.
The next-earliest match is from a 1770 translation of Nicolas Framéry, Memoirs of the Marquis de St. Forlaix:
I have reproaches to make you, Madam ; I desired you to send me Corsange at the beginning of the winter, and I have yet heard nothing of him. I have every reason to wish him with me. He is eighteen, and passes his time with you in the most shameful inactivity. This may be pleasing to you, but it covers me with confusion.
And the next is from an early (1773) translation of Jean-Jacques Rousseau's Emile:
I foresee how much my readers will be surprised to find I have attended my pupil throughout the whole first age of life, without once speaking to him of religion. He hardly knows, at fifteen years of age, whether or not he hath a soul ; and perhaps it will not be time to inform him of it when he is eighteen ; for if he learns it too soon, he runs a risk of never knowing it at all.
Each of these early examples are from a period when the vast majority of allusions to age include "years of age" after the number or "the age of" before it, and it is possible to find special circumstances to justify their departing from this usual pattern. In The Maid's Tragedy, the authors are working in metrical lines and have good reason not to prolong the wording or throw off the meter. In the translation of Framéry's epistolary novel, we may be seeing not an English form but a French one, retained out of fidelity to the French original despite it possible strangeness to the contemporaneous English ear. And in the Emile translation—aside from the same possibility that the wording simply reflects the French original—we have a situation in which Rousseau has already used "fifteen years of age" earlier in the sentence and certainly expects the force of the earlier "years of age" to carry over to the unadorned "he is eighteen."
But even so, we have clear examples where the hearer or reader is supposed to understand "he is twelve" and "he is eighteen" to mean, respectively, "he is twelve years of age" and "he is eighteen years of age." The modern (since 1800) expansion of this expectation to cases where no such ulterior factors are in play simply reflects the growing recognition that "years of age" did not have to be stated explicitly in order for the full idea to be correctly understood.
My examination of early published instances where a person is identified as being twelve, eighteen, twenty-five, thirty-two, or forty-five without the following words "years old" or "years of age" is intentionally scattershot because exhaustive research would be exhausting. But the results suggest that the truncated form was used at least as far back as 1619 and that it was well established (and thus had "become okay") by the early twentieth century.
Shortened forms tend to arise when the longer form is so commonplace that people don't need to hear the entire expression to understand all that is meant. Presumably if people talked about the age of their houses and furniture with the same regularity that they talk about the age of their relatives and pets, they would be much more inclined to say things like "My house is 25" and "This desk is 88." The absence in many cases of precisely known dates of origin for houses and furniture also discourages precision in identifying their age; and if you do know the year when your house was built, you are much more likely to express it as "It was built in 1924" than as "It is 92 years old" (which requires you to update the correct number annually).
