Without using the word "tall", we can say "I am five nine". Can I say "I am five nine tall" as well, or do I have to say "I am five feet nine inches tall" if I'm using the word "tall"?
If you were speaking to someone accustomed to measuring height in feet and inches, you would be perfectly well understood if you said:
I am five nine tall.
If you had been discussing your height, you would be more likely just to say:
I am five nine
I am five foot nine
omitting the word tall.
However, if you were speaking to someone who had always worked with the metric system of measures, you would need to spell it out to be understood.
I am five feet (or foot) nine inches tall
It really comes down to a matter of context.
I believe that some Americans measure their height in centimeters
“I'm a hundred and seventy-five [[centimeters] tall].”
However, omitting "centimeters" and/or "tall" presents a problem, is the speaker talking about their weight or their height?
“I'm a hundred and seventy-five pounds.”
When using feet and inches, I would suggest saying
I'm five foot nine
By inserting "foot" between the two numbers, it's clear that the speaker is referring to their height.
The adjective tall functions as the main predicative in a. but in c. "tall" is allowed to be omitted because 25-inch is attributive and modifies the noun man.
a. General Tom Thumb was twenty-five inches tall.
b. General Tom Thumb was a 25-inch tall man.
c. General Tom Thumb was a 25-inch man.
The OP's height measurement can also be written as
d. I am a 5ft 9 tall man/woman/person
e. I am 5ft 9
You could say it, but it's not the most conventional way to phrase such a statement.
As you wrote, it's quite common and natural for native English speakers in the US (and I assume, also for native English speakers in the UK) to simply say,
in conversation. No special context is required. If you simply said it either way (either that, or I'm five-nine tall) to someone while walking down the street with them, they would most likely understand what you mean and not feel anything strange or puzzling about it.
It's one of those things where, if we (native or highly proficient speakers) are asked to focus on it, it may sound odd or "wrong", but if someone said it to us, we'd likely understand it without noticing that it's unconventional.
Context is always important, however, and the factors that influence it are complex and innumerable.
Google search on he's five-ten shows many examples of this (five-nine does too, but five-ten shows more at the top of the search for whatever reason). I don't see, on a brief scan, examples of I'm/he's five nine/ten tall.
One certainly can say,
I'm five-nine tall
and I'm sure that people do, at least on occasion, but it sounds somewhat strange or unusual to me, and when it's said in conversation, I think it would just be an example of the kind of dysfluency that characterizes speech. (We only imagine that people speak using a "grammar" that appears in grammar books or in prose. If you look at transcripts of most real speech, you'll find that a complete, "paradigmatic" sentence is a relatively rare exception.
In short: not normally, but given the reality of the way people actually use language, I'm sure it's produced at times, and no one could authoritatively say that it's "wrong".
If someone wanted to try to get "technical", I think the grammar of such an utterance could be described as an example of ellipsis (predictable elements of a structure can often be left out):
I'm five [foot] nine [inches] tall.
Note that this does not make I'm five nine ungrammatical. Rather, it would just be an "explanation" that may make some people feel more comfortable that it represents something they can understand from a non-linguistic (i.e. novice) mindset.
For a more satisfactory answer, you might need to explain more about why you are asking, or what you mean by "Can I say . . .?" I suspect you mean something like, "Do speakers of English acceptably say this, at least informally?" And I give this answer in alignment with this sense.