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 "equal opportunity employer". They may have a "mission statement" 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.

This will depend on the institution. This one in particular states that they do not discriminate on the basis of religion.

From http://www.seattleu.edu/policies/nondiscrimination/

Seattle University does not discriminate on the basis of race, color, religion, sex, national origin, age, disability, marital status, sexual orientation, gender identity, political ideology or status as a Vietnam-era or special disabled veteran in the administration of any of its education policies, admission policies, scholarship and loan programs, athletics, and other school-administered policies and programs, or in its employment related policies and practices.

There are some exceptions for religious organizations but it may depend on the state laws and/or the particular job.

You may want to look up what the catholic church says about its universities in order to form an opinion whether you agree with their views before applying: Apostolic Constitution of the Supreme Pontiff John Paul II on Catholic Universities
With a work contract with catholic church you agree with their ethical standards. Depending on close your personal ethics is to theirs, this may be anywhere from advantageous (it may be easier to deny conducting certain research because of ethical objections) to a huge source of conflicts (the typical topics, see below).

AFAIK, the requirements they put on applicants will depend on the faculty and on the job, in particular for catholic theology restrictions may apply that do not apply for other faculties (IMHO quite naturally).

In Germany, churches (i.e. institutions that have an official status of being religious community) are allowed to ask much more duty of loyality from their employees than normal employers. In practice most conflicts about that arise from catholic etics, e.g. employees of the catholic church may loose their job if they try to remarry after a separation (because they cannot be considered leading an exemplary life according to catholic ethics). Also, they may be homosexual, but you may not practice homosexuality (catholic church). And you may not leave the respective church (if you are member to to begin with).
This tends to me more pronounced for employees in the "core church businees" (theological staff, parishes), that educate (kindergardeners, school teachers) or have a publicly visible position.
Another practical consequence is that the employees of churches are often paid lower wages compared to "normal" employers as working there is also considered service (in the christian religious sense).