In many fantasy worlds, each of the various races has their own religious beliefs, and very few cases of characters that are of a certain race, but do not share that race's religious beliefs exist. Is it feasible for various races to each believe in different religions, even in mixed-race areas? What factors would make it more believable?
Yes and no. This is actually a really cool question due in no small part to my facination with how people internalize philosophy. To begin with, we need to consider the different types of foundational beliefs (and I'll include some "non-realistic" just for fun).
BACKGROUND INFORMATION: SKIP TO TL:DR IF YOU JUST WANT THE ANSWER
By the way, these are my own classifications and are by no means meant to be authoritative. I have no doubt that professionals can do a much better job.
Please note that most belief systems have as their foundation the desire to overcome fear. This creates a universality of belief that makes cross-pollination between belief systems fairly simple. Also, most belief systems are highly symbolic, anthropomorphising the ideas and things being worshipped.
Natural: worship of simple natural things such as celestial bodies, weather, and seasons. Nature worship is often tribal in nature, isolated with unique names given to common elements that all people experience. As participants become more self-aware, nature worship often evolves into emotion worship.
Emotional: worship of "human motivation" such as love, hate, anger, etc. This kind of worship usually evolves into the worship of "human inerrelationships" such as war, passion, and deception or the worship of the "human condition" such as birth and death. Emotional worship is often the consequence of greater insight into onself and the interrelationships one has with others. As groups begin to come into contact with one another and communication increases, emotional worship tends to evolve into either ancestor or literal worship.
Ancestral: worship of ancestors, an extension of the idea that with age comes wisdom, which can be passed on to later generations.
Literal: worship of an actual being separate from our world and experience. Often a "super being" believed to oversee and supercede all things. Literal worship tends to evolve from the basic need for "my god to be better than your god." The being simplifies the pantheon of belief while encouraging the authority and power of the being for the benefit of the people.
Fantastic: worship of beings that represent the dispersal of magic in its many forms. This is more than the miraculous, in which the worshipped god applies the solution. It is the means whereby the worshipper can apply the solution at will. Indeed, the worshipper is not simply authorized (as would be the case with a priesthood), but is him- or herself empowered (let in on the secret, access granted so long as worship continues).
Pantheons become more complex as duties and responsibilities of "heavenly management" requires (or desires) greater involvement. This is most notable in warrior beliefs where "lieutenants" are needed to coordinate specific roles or in beliefs that have become patri- or matrilineal as people "make god in their own image" by creating a familial hierarchy.
The greater the isolation, the longer a belief will retain its original expression. As the number of people grow, so does insight into oneself and one's relationships with the world and with others. As peoples grow, once isolated groups begin to mingle. With these changes come the possibility of original belief changing or dispelling.
Skepticism: challenges occur when the worshipped idea or being cannot produce the desired miracle, or when one group of followers challenge the veracity of another. Skepticism is a basic questioning of truth. What is real? What should I believe in? While skeptics are often the challenging force, it is usually the individual (who is forced to look inward and prove the belief to oneself) who brings about change.
Tolerance: challenges occur when the worshipped idea or being of another group of people is seen as antithetical to the woship or culture of the referent group. A tolerant society is one where the worship of many gods is permissible (if not entirely desirable). An intolerant society may act with violence to protect what it sees as its divine purity and purpose, cleansing others to promote its own superiority.
Schism: challenges that occur within a society that don't change fundamental belief but do change critical details about that belief such that two peoples, each believing in one side of the schism, are no longer compatible despite the common belief (and often the common culture).
TL:DR; The answer you've all been waiting for....
The answer is "yes, if and only if you have developed a culture with either (a) tolerance or (b) isolation." Tolerance assumes a middle ground in that the religious foundations between the cultures are neither too similar (we all worship a sun god, but some worship hes, other shes, by different names, which is "correct?") nor too disparate (they're natives, they still worship the sun, we all know that Crom is the one true god!).
Isolation is needed if there is difficulty relating to the various cultures. Humans, at least, generally do not trust what they do not understand and rarely trust people who are "not like me." Even separate languages have been the cause of distrust.
Which means the ultimate answer is often "no," unless you have contrived a means that it can be otherwise.
If you look at most religions you tend to find common themes. Man is either created in the image of god(s) or descended from them. With that in mind it makes sense that elves, dwarves and humans would all worship different gods.
If your gods are more generic (God of Fire, God of the Sea) with no race or racial connection then it is more likely you will have cross species religions.
Short answer, yes. But it will not be free, the society pays a price.
Creation myths are old and in the times now past populations were low and often not in touch with one another. In a world with more than just humans, this contact could have been even less or non-existent between the races. For example, the different races could have easily begun in geographically distant areas. Or inaccessible areas. Then, after religions began, moved out from there.
Most religions start with/are based on a creation myth. The crossover we see in Earth's religions most likely comes from early contact/conquest. Minimizing that in your world's history will also help strengthen arguments for no cross-over in belief systems.
As populations spread, after a point in which each race has solidified their system(s), contact is a little less likely to result in contamination of beliefs. But not totally.
You will need reasons for that to continue. Humans as an example, tend to question life and the universe. They look for meaning. Should they become disenchanted with their current system of beliefs they may seek something "better". Subjectively better but that's the point.
So, to continue, there must be a social mechanism that promotes isolation if the belief systems are to remain "pure". That can quickly get into the realm of totalitarianism unless there is a signification geological barrier to promote the isolation.
Looking at Earth, we don't have such barriers for a large percentage of the population. As a result, we see more totalitarian mindsets used to combat the spread of ideas. For example, see North Korea.
So, yes you can make a world like you want. But you have to reasons as to why it's that way. Those reasons will tend to shape the world and its peoples.
First, we already have Judaism as your example.
According to Wikipedia, Judaism has 3000 years worth of history. The Jewish diaspora/exile from almost 2000 years ago to various parts of the world did not make them lose faith in their God, or mixing with other faith.
Just make it that it won't make sense if other races convert to other race's religion.
It will not make sense if cave dweller dwarf worship "Fay", the goddess of all flying creatures, which is worshiped by the avian race.
Make the religion be connected to the physical features of each race.