If Dr does not take the period(.) because the last alphabet of Doctor is the same with the contracted form Dr, why does Saint take period(.) after it(Saint and St.)?

Oh, great question.

I could argue that St. shouldn't get one, but I don't feel comfortable with that as an answer. The best I can offer is that it's an exception.

I should also point out that this isn't really a strict rule anyhow. It's just a tidy guideline.

Full stops/periods in contractions