Can a person who worships Hashem, follows all commandments, keeps the Sabbath holy and follows Jewish tradition but is not a born Jew wear a Jewish head covering and tell people he is Jewish by faith?
In Jewish Law, there is a prohibition known as “geneivas daas”, which is literally translated as stealing one’s mind/heart, and refers to various forms of deceit and giving false impressions.
Although the Torah does not present the concept as a formal prohibition, many authorities maintain that geneivas daas involves a full Torah transgression. According to the Ritva (Chullin 94a), the source of the prohibition is the transgression of theft. Although one might think that theft is limited to stealing money or goods, the Ritva (citing Tosafos) explains that it applies even to theft of the “heart.”
The Semag (Negative Prohibition 155) and the Shulchan Aruch Harav (Onaah 11) likewise state that the prohibition of deception is a Torah law. The Semak (262), however, writes that the prohibition is rabbinic, and this is also the opinion of the Bach (Choshen Mishpat 228).
Being that the “Noahide Code” (which applies to all non-Jews) includes a prohibition against “stealing”, the question has been raised if non-Jews are also prohibited from “geneivas daas”, since this can be viewed as a subset of stealing.
[See Rabbi Shimon Sofer, Shut Hitorerut Teshuva, 1:118) here]
If in fact a Noahide is prohibited from “geneivas daas”, then a non-Jew pretentiously masquerading as a bona fide Jew, would be guilty of “stealing” the minds and hearts of the people he comes into contact with.
Yes, he can wear a head covering and tell people he's Jewish, but it wouldn't be true. But I've never heard of any requirement on gentiles to tell the truth.
However, this lie could be damaging to whatever Jews he tells it to, so he should avoid it. For example, they may count him toward the required number of Jews present for a communal prayer service, when non-Jews don't count for that.