“Her sympathy was ours” is ambiguous between two meanings:
“her sympathy was [the same as] our sympathy”, that is, she was sympathetic to the same things that we were sympathetic to (whatever they were);
“her sympathy belonged to us”, that is, she was sympathetic with us.
In both readings, “we” and “us” refer to the narrator, Victor Frankenstein, and his friend Henry Clerval. In the second reading, “sympathy” is being used in this sense:
sympathy, n. 3a. Conformity of feelings, inclinations, or temperament, which makes persons agreeable to each other; community of feeling; harmony of disposition.
Oxford English Dictionary
and not in sense 3c, “the quality or state of being thus affected by the suffering or sorrow of another”.
The remainder of the sentence, which says that her smile, voice, and eyes “were ever there to bless and animate us”, inclines us towards the second meaning: if these attributes were always there for us, then presumably her sympathy was always there with us too.
Clearly the whole passage is making use of religious images: Elizabeth is described as having a “saintly soul” which shines like a “shrine-dedicated lamp”; she has “celestial [heavenly] eyes”; she “blesses” us, and so on.
However, I don’t find any specific Christian reference in “her sympathy was ours”. The word “sympathy” in the sense used here is an early modern coinage, and is not found in the King James Version of the Bible, which uses “pity” or “compassion” instead. Moreover, the suggestion, made in comments, that the Catholic doctrine of intercession of the saints is being alluded to, seems quite implausible:
Intercession is a process in which the believer asks the saint to influence God on their behalf, but Frankenstein does not seem to be asking anything of Elizabeth in this passage.
Mary Shelley was not especially sympathetic to Catholicism. For example, she wrote:
we are disgusted by the Catholics setting the corruption, profligacy and imbecility of Papal Rome, in comparison with her ancient glories.
Mary Shelley (1829). ‘Modern Italy’. In The Westminster Review, number XXI.