The First Amendment essentially prohibits the establishment of a list of approved vs. not-approved religions. There is a large body of First Amendment case law that relates to claimed religious beliefs, and the restriction that the government cannot prohibit a person from exercising their religious beliefs. The courts therefore avoid bright-line answers to the question.
You can look at Friedman v. Southern Cal. Permanente for an example where a court found that a belief is not a religious belief, finding that veganism is not a "religious creed" within the meaning of the California Fair Employment and Housing Act. Plaintiff was told to get a mumps vaccination (which involves chicken embryo) as a condition of employment, refused, and the offer of employment was refused. The law, §12940a makes it unlawful "because of the religious creed of any person, to refuse to hire or employ the person...". Subdivision (l) says that
Religious belief or observance, as used in this section, includes, but
is not limited to, observance of a Sabbath or other religious holy day
or days, reasonable time necessary for travel prior and subsequent to
a religious observance, and religious dress practice and religious
grooming practice as described in subdivision (q) of Section 12926.
§12296(q) further contributes a definition of religious terms:
“Religious creed,” “religion,” “religious observance,” “religious
belief,” and “creed” include all aspects of religious belief,
observance, and practice, including religious dress and grooming
practices. “Religious dress practice” shall be construed broadly to
include the wearing or carrying of religious clothing, head or face
coverings, jewelry, artifacts, and any other item that is part of an
individual observing a religious creed. “Religious grooming practice”
shall be construed broadly to include all forms of head, facial, and
body hair that are part of an individual observing a religious creed.
The Fair Employment and Housing Commission then created a regulation California Code of Regulations, title 2, section 7293.1 that defines "religious creed"
‘Religious creed’ includes any traditionally recognized religion as
well as beliefs, observations, or practices which an individual
sincerely holds and which occupy in his or her life a place of
importance parallel to that of traditionally recognized religions.
The court notes that
religious creed extends beyond traditionally recognized religions to
encompass beliefs, observations, or practices occupying a parallel
place of importance “to that of traditionally recognized religions” in
an individual's life.
The court then points to the leading cases from the Supreme Court related to deciding what a religion is: US v. Seeger, 380 U.S. 163 and Welsh v. US, 398 U.S. 333. And the court points out that government agencies are granted wide latitude in interpreting enabling legislation (under which the government might allow or disallow an action).
It us noted that as of 2002, there were no California cases deciding what constitutes a religious creed w.r.t. FEHA and that regulation. The court then recited various statements from other California cases regarding the characterization of a religion – the main relevant point is that a religion does not have to be theistic to be a "religion". The court also reviews federal employment discrimination laws, and again considers the difference between traditional and non-traditional religions. There are many snippets in the review of the law section saying things such as that
the court should find beliefs to be a religion if they ‘occupy the
same place in the life of the [individual] as an orthodox belief in
God holds in the life of one clearly qualified.’
We can now move to the question in §6, Is Veganism a Religious Creed For Purposes of the FEHA. This court states that
The test we apply is that set forth in Judge Adams's concurring
opinion in Malnak which has been adopted by the Third, Eighth, Ninth,
and Tenth Circuits
which they say
presents the best objective method for answering the question whether
a belief plays the role of a religion and functions as such in an
individual's life
At this point, we can't reasonably guess how some individual would object on religious grounds to the vaccination mandate, but we do at least know what aspects of the law would be relevant and where the "rules" are laid down. Until someone actually alleges that they have a religious belief that prohibits vaccination, we can't analyze those arguments. The gist of the "veganism is not a religion" is it is too narrow a belief to constitute a religious belief. The court found that did not find that it is a "belief system (which) addresses fundamental or ultimate questions", and that it does not a address fundamental questions such as "the meaning of human existence; the purpose of life; theories of humankind's nature or its place in the universe; matters of human life and death; or the exercise of faith".