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For this simple cipher, this will also immediately reveal the key.
Yes, that's exactly what format-preserving encryption does.
This is also easily detectable, although recovering the key may be more — possibly much more — difficult.) In fact, the methods to use for breaking your cipher are basically the same as one would use to break a simple Caesar cipher.
Both are simple additive monoalphabetic substitution ciphers; the only difference is that, in your case, the "alphabet" consists of months rather than of letters.
Then again, in the real world, your data is never just a bunch of random dates.
Rather, the dates will be dates something, and might be associated with other data (names, descriptions, etc.) that could reveal any manipulation under close inspection. Some months see noticeably more births than others (for example, there's a notable peak in births in September, about 9 months after the Christmas holiday season), and there are also significant weekly trends (presumably because C-sections and induced labor are not usually scheduled for weekends) and sharp peaks and troughs around specific holidays (presumably for similar reasons).