We have no indication, as far as I know, that birthdays are celebrated, or even remembered, in the Star Wars universe. There are Earth cultures that don’t care about birthdays, so it’s not in any way improbable.
Besides, what if they noticed? I shared a birthday with a girl I went to high school with, but when I found that out I didn’t think “We’re probably twins that were separated and adopted away because our father went nuts and betrayed his friends”.
I’m pretty sure R2 did NOT have his memory wiped, only C3PO.
Luke: Wanna make out?
(I’m sorry. I just found that disturbing.)
God, I hate when that happens.
But ya gotta wonder, what would have happened had their midichlorians co-mingled.
c3p0 was anakin’s creation. after the death of his mother c3p0 went with anakin.
r2d2 part of padme’s ship. after his heroics in ep.i he was allowed to come along on missions. he ended up with anakin and had a part in helping him destroy the droid army.
c3p0 and r2d2 were exchanged as gifts when padme and anakin secretly married. r2d2 worked better with a jedi who was flying around saving the universe.
c3p0 worked better with a senator who had to meet with many different people.
r2d2 and c3p0 met up at mustafar and flew away with obi-wan and padme. when padme died the droids went to leia (oldest child) and into the organa household.
obi-wan had contact with both droids but never owned them.
now if the question was: “do you know these droids?” there may have been a different answer.
And when did she know it?
Someone had to. 
Also, adopted children often don’t know their birthday. A friend of mine growing up always celebrated his “adoption day” instead, which is when he became a part of the family, and became a child to the people he considered his parents. He always thought of it as his birthday until he was old enough to be told the difference, and whenever anyone would ask his birthday, he’d just give them his adoption date.
Not sure if this is still The Done Thing, but there you have it.
Also: Lucas is a hack.
Didn’t Luke and Leia have a sexual relationship in “Splinter of the Mind’s Eye?” I had thought that Lukas gave his blessing to the general plot lines of all the novels, and they were considered authoritative, but this one seems to have been “overtaken by events.” Is it still considered canon? One of my friends is a Star Wars fan and had never heard of it. I gave her my old copy and she was shocked, to put it mildly.
Luke and Leia did not have sexual intercourse in Splinter of the Mind’s Eye, but I think they kissed. Alan Dean Foster was clearly keeping the sexual tension alive for the as-yet-unnamed second movie.
Or he coulda been sitting around thinking to himself “Hmm… how should I tell him this… ah well, I’ll just tell him later. She’s way out of his league anyways, and no doubt he’ll meet some cute rebel chica and hook up. It’s not like I’m gonna have to sacrifice myself so they can escape. Oh shi–” 
That’s likely, considering that his entire strategy in re: Luke is completely retarded.
"Let’s see, I’ve got the son of an incredibly powerful Sith Lord here, who’s likely to become extremely powerful himself one day, and if that happens he’ll be a threat to the Emperor. I guess what I’ll do is take him to his father’s homeworld and turn him over to his uncle by marriage, who isn’t all that thrilled with Jedi, by the way, thus insuring that he’ll get no worthwhile training for the first twenty years of his life, and I won’t even take the simple precaution of changing his freakin’ name.
Oh yeah, then I’ll go live in the desert."
“… and talk to ghosts.”
and, as we the audience, are to find out twenty-some-odd years later, commune with the dead?!
Heh, my first simulpost. I’ll change my comment to: “Age forty years while Luke only ages 17.”
Sumbitch came back as a ghost and STILL didn’t warn him.
Wait! I GOT IT!
That is the real reason he sent Luke to Dagobah.
He was pawning the responsibility off on Yoda!
Deleted Scene From Empire Strikes Back:
(Luke is freezing and sees Obi Won on the ice plains)
Luke: One more question, Ben! How can I get Leia to like me?
Obi Won: What? What? I can’t hear you. There is a lot of interference in the force. My signal is breaking up. Ok. If you can hear me, I am going to hang up now. Go to Dagobah. Ask Yoda any questions…
Luke… that is not the ass that you wish to tap.
It’s also possible that Obi Wan knew he was a putz, and was therefore afraid to tell Luke anything. He doesn’t really tell the kid anything other than general information about the Jedi and some basic force usage, and probably intended to ship him off to Yoda even if he didn’t get killed. He actually seemed kind of reluctant to even go that far.
Considering how his last student turned out it was probably a smart move.
or better yet
Then how come R2 forgot how to fly?