Throughout Jewish history, there have been people who have been known as "philo-Semites", those who love Jews and Jewish culture. We find it in our ancient past as well as in modern times (e.g., the various Jewish themed Klezmer and Yiddish festivals in Poland). There are many who look positively upon this phenomenon and see it as an opportunity for Jews to appreciate their Judaism more and, at the same time, bring additional positive awareness of Jews and Judaism to the world at large. On the other hand, there are those who see it as troubling because it blurs the lines between those who are halachic Jews and those who are not.
In answering your question, I would view these philo-Semites in a positive light but with several caveats. They should not portray themselves as Jews to others or to themselves. They cannot receive ritual honors or participate in other specific Jewish rituals which are accorded only to those who are halachically Jewish. There is a difference between those who are Jewish and those who have not undergone the proper ritual procedures for conversion. Nevertheless, if these philo-Semites accept the limitations placed on them by not passing themselves off as halachic Jews, it perhaps can be a positive development in promoting Judaism and Jewish culture to the world.
Your question has an answer that appears to me to change over time. In Biblical times we are told that there were ‘hangers on’, and ‘camp-followers’, and that when the Hebrews left Egypt, they went as a ‘mixed multitude’ – i.e., not all Hebrews. The Tanakh (Jewish bible) speaks of Gerei Toshav (persons who are not part of the people but who lived among them and followed the practices of the Hebrews/Israelites). Ruth, the Moabite, could be considered to be in this number (though it is generally accepted that her assertion to Naomi served as a full formal conversion).
We are told that when the exiles returned from Babylon, to minimize the problems that had arisen, Ezra and Nehemiah ordered that they had to separate out those who were not actually part of the Israelites and send them away. In most other times and places earlier in Jewish history, there were those who lived among the people, but were not actually ‘officially’ a part of them. So in ancient times, there were people who fit the description in your question; that practice was seemingly stopped in the 535 B.C.E. time frame.
So, as the saying goes, that was then, and this is now.
Today, in accordance with the ruling of Ezra and Nehemiah, it is the general understanding that if one wishes to become a Jew, they must complete a formal conversion process and be accepted. Anything else, or less, does not count; the standards are more formal, and the prerquisites are more specific.
The answer to your question seems to me to be that those who are not Jews should not undertake to practice, act, or perform as one. It is disrespectful and a usurpation and diminution of the rites and rituals which are not theirs and which they mimic (just as it would be for other faiths).
Because there are still those who wish to live in overall accordance with a Jewish outlook and follow some Jewish practices (but not follow a fully Jewish life), there is an alternative that is available. This is not a conversion to Judaism, but it also fulfills many of the ethical and moral teachings, without imposing all of the mitzvot (commandments) on the person who chooses this path. It is well defined, and readily accessible.
There is, in fact, an entire body of literature and instructions for these persons, and there are communities of them around the world. They are called B’nai Noah (children of Noah); they consist of those who follow the Noahide laws. These laws are described as the seven commandments which were applicable to Noah, and therefore to all human beings. These laws do not extend to or encompass the 613 mitzvot that make up Judaism. B'nai Noah are not Jews, and do not practice Judaism - they do, however, live in a manner that fulfills the common commandments that apply to all humans, including Jews.
The Noahide laws and the information are readily available online; for brevity I will not try to summarize them here. This is the appropriate path for a non-Jew who wishes to take on some part of Judaism as their practice and spiritual path, but not to become Jewish.
So, to answer your question specifically, I would say that one who is not a Jew should not undertake to practice as a Jew for good and sufficient reasons (and equally, should not call themselves a Jew); just as one who is not a member of another religion should not undertake to take on the practices of that religion. It is simply a matter of respect and truthfulness. They have an alterative available to them, or they may follow any other religion that also recognizes these seven universal laws, and live a good and fulfilling life in accordance with G-d's teaching.
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