In some contexts it could also refer to shoe or shirt size, as they are also commonly called "numer" instead of "rozmiar" (size), but it should be then very clear what we are talking about, because to avoid ambiguity, people would rather say "Nie znam jego numeru buta" [I do not know his shoe size] than "Nie znam jego numeru". Technically, it can also refer to other numbers, as bank account number or PESEL, (a national personal identification number, assigned to everyone at one's birth), but then also in most cases it would be stated which number we are talking aabout, f.ex. "Jaki jest jego PESEL?" -- "Nie znam jego numeru." [What is his PESEL? -- I do not know his number.]
"znam" is like "I am familar with it", even if 'being familiar with a number' sounds strange. "wiem" is like "I have this knowledge".
In most situations this simple rule will help: "znać" = "to know X", "wiedzieć" = "to know about X", "to know, that X...".
again the stress is wrong: it sounds like numeru is stressed on the first syllable
"cyfra" = figure, digit. It is a single character, used to write numbers, exactly as letters are used to write words. It does not have a mathematical value itself, but is used to write expressions that mean numbers. In decimal system there are exactly 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 - in other numeral systems there may be more or less - eg. in hexadecimal: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F; in Roman numbers: I, V, X, L, C, D, M. See also: https://sjp.pwn.pl/szukaj/cyfra.html
"liczba" = number, figure, amount. In maths it is a basic concept that represents a given value. In everyday life it is used to express values, amounts and quantities. In grammar it may be "singular" or "plural". See also: https://sjp.pwn.pl/szukaj/liczba.html
"numer" = no., number, issue, also party piece. It is a set of digits that does not represent a "value", but is rather an index or identifier of an object in a series or group of similar objects that are used for something: eg. number of the building in the street, postal code, telephone number, bank account number, issue of a periodic magazine - these "numbers" do not represent values, but either mark one object in a sequence of objects (magazine issue, number of the building in the street), or serve for identification purposes only (telephone number, postal code). In the other meaning, it is a single "act" in a show, eg. one performance of many of them, during a show in a circus. See also: https://sjp.pwn.pl/szukaj/numer.html
Would wiem work in this case or would it be an improper use of the word.
No, "wiem" does not quite work here. If you really want to use it, it should be "Wiem, jaki (on) ma numer", but that form would be a bit strange (unless this number is classified or knowing the number is somewhat uncommon - in such a case, this is the best way to say it).
Here is an article about Polish translations of the verb "to know": https://forum.duolingo.com/comment/25535826