your birthday as far as I know does not impact the character, but the guardian does, if/when picking a god/goddess pay attention to the elemental attributes listed under the god/goddesses description. You will notice these numbers differ from each god. Some boost your fire element more while others your water ect. Depending on what your role/job will be some of these gods boost needed attributes more than others.
For example a black mage's primary attack is Flare which is a fire elemental base spell. Choosing a guardian with a higher base fire element will help boost this spells output.
That is just one example but it works the same for all jobs.
I should note though the base number are so minor of a boost, it really doesn't matter what you choose it isn't going to make or break your character in anyway.
Some websites say its a elemental attribute bonus, some say it is a resist only bonus, so I don't know which one is right. I do know that it isn't really that important, and I would just pick the one you think looks good or sounds the coolest to you.
The character creation says is resistances, but either way the differences are so small it doesn't make a noticable difference. Even min/maxers would have a hard time with determining it.
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your birthday as far as I know does not impact the character, but the guardian does, if/when picking a god/goddess pay attention to the elemental attributes listed under the god/goddesses description. You will notice these numbers differ from each god. Some boost your fire element more while others your water ect. Depending on what your role/job will be some of these gods boost needed attributes more than others.
For example a black mage's primary attack is Flare which is a fire elemental base spell. Choosing a guardian with a higher base fire element will help boost this spells output.
That is just one example but it works the same for all jobs.
I should note though the base number are so minor of a boost, it really doesn't matter what you choose it isn't going to make or break your character in anyway.