What is the significance of Cassius saying "This is my birthday, as this very day was Cassius born" in Act 5, Scene 1, Line 73 of Julius Caesar?

There is probably no special significance to Cassius's reflection that this is his birthday. Shakespeare was drawing on Plutarch's lives of Caesar. Brutus, and Mark Antony for his play. He frequently inserted little comments from Plutarch directly into his script just to capture the feeling of the time period. According to the "Introduction" to Julius Caesar in The Pelican Shakespeare:

Never before has he relied so heavily on one source--in this case, Thomas North's 1579 translation of Jacques Amyot's 1559 French version of a Latin translation of Plutarch's Lives of the Noble Grecians and Romans. . . . At certain junctures, the playwright turns North's prose directly into iambic pentameter, barely changing a word.

The effect of Shakespeare's play, even though it is written in English and in iambic pentameter, is to carry the viewer back in time to ancient Rome. Shakespeare creates an hypnotic spell. The viewer becomes a time traveler. Most modern readers' impression of the characters and events connected with Caesar's assassination and its aftermath are derived from Shakespeare rather than from history books. This includes Mark Antony's famous funeral oration, which is probably nothing like what the real Mark Antony said to the mob but seems somehow to be what he should have said.

Plutarch probably quoted Cassius as saying that it was his birthday because Plutarch was that kind of conscientious historian. His "Lives" are full of little details which he picked up from his reading, or from other scholars, or from hearsay. Shakespeare, of course, had to invent most of the dialogue, but he was probably happy to find in Plutarch a line or two of dialogue which he could incorporate verbatim into one of his plays.

What does Cassius's "this is my birthday" speech foreshadow in Julius Caesar?

Cassius describes omens of their doom at Philippi, including fallen eagles.

Cassius gets superstitious on his birthday.  He has always claimed that there is no such thing as omens.  Now, facing a battle that he does not feel he can win against what he thinks is a stronger foe, Cassius decides that the omens are against them.

And partly credit things that do presage.
Coming from Sardis, on our former ensign
Two mighty eagles fell, and there they perch'd,
Gorging and feeding from our soldiers' hands;
Who to Philippi here consorted us … (Act 5, Scene 1)

For Cassius, the thought of being born and dying on the same day is too much for him.  It makes him depressed to the point of losing hope.  Cassius loses his fighting spirit.  He believes in bad omens, and he thinks he is doomed.

There is also a strange irony here that Shakespeare wants to make sure that the audience does not miss.  A birthday is supposed to be a celebration of life, not a harbinger of death.  Yet people also look at their lives on their birthdays.  For Cassius, it is a chance to take stock and realize that he failed to accomplish everything he set out to do, and now he may die on his birthday.  Cassius's lack of faith in himself stems from the fact that it is his birthday.  It is a self-fulfilling prophecy. 

When Messala tells him not to believe it, Cassius says he believes it “partly.”  He tells Brutus that they are about to face the unknown.

Now, most noble Brutus,
The gods to-day stand friendly, that we may,
Lovers in peace, lead on our days to age!
But since the affairs of men rest still incertain,
Let's reason with the worst that may befall. (Act 5, Scene 1)

Brutus and Cassius agree that they will not be taken.  They do not want to let Antony and Octavius march them through the streets of Rome in triumph.  They decide the only way to avoid such indignity is to kill themselves.  This is how Cassius ends up committing suicide on his birthday.

With Cassius gone, the omens prove true.  There is no way Brutus can carry on without him.  Cassius was too ready to commit suicide.  He was so depressed and spooked that he mistook a victory celebration for a capture.  Brutus is hardly better off.  He decides a noble death is easier than fighting.