Are the odds of someone dying on the same date of the same illness as someone else 365 to 1? How are odds figured?

It's really quite simple but you do have to clarify the definition of same date and same illness. eg is the same day of the same year or the same day but possibly differing years. The latter gives 365 to 1 ignoring the illness bit and leap years. 

To bring in illness, you have to bring in mortality tables split by illness (effectively defining the sameness of the illnesses). Then you merely have to multiply up the relevant probabilities and sum up by date. 

A spreadsheet should deal with the computational complexity. The odds would therefore differ by person. 

The odds will vary according to the known facts at the time. Before Senator McCain passed away the chances of them dying on the same day of the year would approximate closely to 365 to 1. After then until TEd's death his chances of dying on 25 August would approximate less closely to 365 to 1 expecially after diagnosis. After both deaths all bets are off. 

To take into account cause of death you would factor in the mortality table for death by brain cancer (or each cause before McCain's death) applied to the relevant calendar dates. Clearly the applicable mortality would rise sharply to become calendar sensitive after diagnosis.