This question popped in our exam in which you had to choose which particle was
right. The question was:

> 誕生日___母に花をあげました。

with に or まで as the options. まで didn't make sense for me and I don't know if
going with に is possible.
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grammar particle-に
Is it possible to have two words with the same particle in a sentence?
Yes it's perfectly possible to use the same particle twice or more in a
sentence and even in the same clause, and it is very common.

There are some restrictions e.g., I don't think you can have the object marker
を more than once within a single clause, but then を only has one function
(almost). に is a particle with many different functions.

You are correct that the missing particle is に and this does not cause any
ambiguity or difficulty in understanding.

> 誕生日 **に** 母 **に** 花をあげました。  
>  (I) gave flowers **to** mum **on** (her) birthday.
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It is possible to have. に has different meanings after 誕生日 and 母.
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Yes it is possible and repeated use of に in your sentence would be the most
correct and natural way.

誕生日に on my Mom's birthday (implied to be Mom's birthday based on context)

母に to my Mom

花をあげました I gave flowers

There is no rule that a particle cannot be repeated in a sentence and your
sentence can be further extended.

> 私の誕生日に母に花をあげて父にあげて弟にあげておばあさんにあげずに逃げました。
>
> On my birthday, I gave Mom flowers and I gave Dad flowers and I gave my
> little brother flowers and without giving Grandma flowers I ran away.
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answer