Bdtyk

I found this earlier use of the phrase from a political poem in Volume 7 of Punch, 1844, which might indicate a law-enforcement origin:

The OED associates it with the earlier phrase to take measure [of] ("to form an estimate of; to weigh or gauge the abilities or character of, or assess what to expect from"), which dates from the 17th century.

The earliest citation of get/take/have [one's] number is from Dickens' Bleak House, published in 1853:

Whenever a person proclaims to you ‘In worldly matters I'm a child,’ that person is only a crying off from being held accountable, and you have got that person's number, and it's Number One.

I doubt we'll find a definitive "first use", but here is an example from 1915 explicitly making the point that (for some people, at least) the expression was considered "yesterday's slang" even then.

In terms of the semantics, I doubt the origin owes much to actual house address number, telephone number, etc. I think it's a fairly transparent metaphorical usage, indicating that the speaker thinks he's successfully classified the person he's speaking to. And would therefore easily be able to put him into a hypothetical numbered 'pigeonhole', and know where to find him again later.

In most usages today, the expression means something akin to "You can't fool me", meaning the speaker has classified the other person as devious, self-seeking, and untrustworthy. So it's usually derogatory, which it wasn't necessarily in the past.

LATER: Fanciful, perhaps, but I'm quite prepared to believe Dickens's wry observation on the specific case of someone with "Number One" as his number (per @phenry's Answer) could have been influential in causing the expression to shift from positive/neutral understanding of another's primary concerns, to seeing through another's deceit.

In Revelation 13:18 (KJV) we read:

Let him that hath understanding count the number of the beast: for it is the number of a man; and his number is Six hundred threescore and six.

These words are usually thought of as pointing to some specific man who fits into the prophecy in the previous verses. You’ve got his number. It’s 666. Now, if you understand the prophecy, figure out who this man is. One popular interpretation in ancient times was that Nero Caesar was the man whose number is 666.

In ancient Greek and Hebrew, every letter of the alphabet also served as a numeral, and thus had a numerical value. Consequently every word, phrase, or name had a numerical value which was the sum of the letter values. And it was believed that one could learn something about the essence of a god or a man by looking at the number of his name, and comparing it to words and phrases that have the same number. This sort of numerology is called gematria if referring to Hebrew, and isopsephy if referring to Greek.

There is no guarantee that the idiom “I’ve got your number” has its origins in the ancient practice of gematria. But it seems pretty obvious that it might. And likewise for metaphorical language such as “he weighed his words carefully” or “he spoke in measured words”.

Etymonline reports that:

To get or have (someone's) number "have someone figured out" is attested from 1853.

Sadly it doesn't offer any insight as to how the phrase came about, but clearly it can't be related to telephone numbers.

My first-thought also points back to gematria. In this system, each word (thus name) has a numerical value. Correlated with this fact is the ancient practice of summoning (angels/demons).

Demonic signatures have a numerical value. That number derives from the numerological value of the seal used in the conjuration of the demon. By knowing this number, a magician has power over said demon, e.g. to summon them to do one's bidding. The most famous account of this still re-told today Legemeton that Solomon used to summon Asmodeus & build the First Temple in Jerusalem.

This assert of power, to compel a demon to do one's bidding, stems from the concept of True Name. YOu can think of this like using a name not to address the person, but to 'address' their soul or or fundamental, primary, essence.

The concept for this arises from the existence of same number-name correlation for angelic names. These true numerical names are thought to originate from the divine, as a kind of 'programming language' for the Creation.

From all of this later derived the Numerological lore of thinking that a system for calculating a number for a person's name had introspective power into their true nature.

There was BBC timewatch that stated that a 18teenth century shipbuilder had the numbers on workers' clothes and if he saw any slackers they were fired hence,`your number is up.

I have in my possession a book first published in 1927 called "I've Got Your Number," by Doris Webster and Mary Alden Hopkins, which was a self-help of book that used numbers to create a personality profile for men and women. The book was apparently very popular and continued in print till at least 1996. The book provides no explanation of how the profiles are arrived at. There is no way to say for sure if this is the origin of the term or merely an outgrowth of it, but I would suspect it may have had a great deal to do with the popularization of the term.

I have no idea where I've heard this first, last, or even how often - but I've heard it more than a few times as "You've got my number!" as in "You've got me" or "You know my weakness." In that context, it's more or less the same as "Well played, sir."

A speculation on the origin of "I've got your number" is a military origin.

Ballistics, particularly, was a calculated endeavor. I've heard the expression "it's dialed in", or similar, used in relationship to ballistics – and in that context the 'dialing' is the adjustment of guns relative to angle-bearing dials (measures) that aid in precision for accurate shots.

There's an interesting similarity between having a shot 'dialed in' and, possibly, 'having the number' of the target.

Since delving into the enneagram, an ancient wisdom of the personality, I’ve thought thus phrase may come from it. The enneagram assigns numbers 1-9 to the various types.