Are employments in universities affiliated (or named) with a specific religion restricted to people with the same religion?
In my experience, the answer is usually no but occasionally yes.
Note: This answer is specific to the United States.
Most employers in the US are forbidden by law from discriminating on the basis of religion; they have to give equal consideration to candidates of any religion or no religion, and generally they will avoid asking about religion at all.  This includes non-denominational academic employers.
However, religiously affiliated employers (such as churches, religious charities etc), are exempt from this rule, and may use religion as a criterion for employment if they so choose.  This includes religiously affiliated academic institutions.  Each such institution will have its own policy on whether they consider religion and what the criteria are.  These policies will usually be mentioned in job postings.
In my experience, most intentionally choose to not consider religion as a factor, usually giving the justification of creating an inclusive community.  These may be institutions that were founded by a church and may still receive funding from them, but operate mainly as a secular institution, accepting students and faculty without regard to religion.  Job postings will often indicate this with phrases like &quot;equal opportunity employer&quot;.  They may have a &quot;mission statement&quot; with religious wording, and a tradition of religious activities on campus (such as regular church services), but nobody is obligated to participate.
There are others which choose to be more overtly religious.  They may accept students and faculty only if their religious beliefs align with those of the institution.  These are more likely to have formal religious activities on campus, and sometimes strict moral conduct codes.  Their job postings often ask explicitly about the applicant's spiritual beliefs.
The institution's name is not always a good indicator of where they fall in this spectrum.
As one example, Nebraska Wesleyan University describes itself as:

an academic community dedicated to intellectual and personal growth within the context of a liberal arts education and in an environment of Christian concern.

However, they also say:

Nebraska Wesleyan University provides equal educational opportunities to all qualified persons in all areas of university operation, including education and decisions regarding faculty appointment, promotion or tenure, without regard to race, religion, age, sex, creed, color, disability, marital status, national or ethnic origin or sexual orientation.

At the other extreme, Dordt College included this wording in a recent job posting:

The Mathematics, Statistics and Computer Science department at Dordt College has a history of preparing students for graduate school, industry and K-12 education. This preparation is infused with an unashamedly Reformed Christian worldview, in order to prepare students to be global citizens in God’s service.  [...]
We look forward to receiving applications from candidates who wish to join us in [...] teaching mathematics from a Christian perspective in the context of educating the whole person.
Qualified persons committed to a Reformed, Biblical perspective and educational philosophy are encouraged to send a letter of interest and curriculum vitae/resume [...]

Generally, institutions that will only consider applicants of a certain religion will make this clear at the outset.  However, even if not, you will have to decide if the religious environment (or lack thereof) on a particular institution's campus will be comfortable for you.  This is an appropriate  topic to ask about during an interview.