In your phrase "get something for your birthday" it means receive. But keep in mind that 'get' is a word with a lot of meanings. It can also mean to buy ("I'm getting a new pair of sunglasses at the store later today."), or even to become ("I'm getting smart" or "I got good at English this year.")
Even more complex, in colloquial English, particularly in the Southern United States, it can be contracted with "to" in increasingly degenerate ways to mean "must" or "have", as in the phrase, "I have got to go to the mall" (meaning "I must go to the mall" or, more commonly, "I need to go to the mall.") which becomes "I've gotta go to the mall." It can degenerate further by losing the auxiliary verb "have" to become "I gotta go to the mall." This is, of course, grammatically incorrect, but you may hear it in some locales and dialects, especially uneducated ones.
At the end of the day, 'get' is a very complex word. Hope this helps.