Get in the meaning of the past in the sense, but for some reason it is written in the present
No, it's all in the present.
I am married to my best friend means I'm still married to him/her and plan to continue to be married to him/her for some time, possibly for the rest of our lives.
I get to be married to my best friend means I am (lucky enough; strangely enough; or whatever) married to my best friend.
"I get to" means, essentially, "I've somehow managed" or "I'm allowed to."
As in:
"Would you like to accompany me to the concert?"
"Sure, provided I get to listen to the music, too, and not just your gossip."