Sirius probably was going for the sentiment, not strict accuracy.
When Sirius told Harry that the Firebolt was thirteen years worth of birthday presents, he meant it as a way of apologizing for being not in Harry’s life for such a long time. He also gave Harry the Firebolt for Christmas, not his birthday, so it technically wasn’t even a birthday present.
“He got a Firebolt for Christmas,’ said Ron.
‘A Firebolt? No! Seriously? A – a real Firebolt?”
- Harry Potter and the Prisoner of Azkaban, Chapter 12 (The Patronus)
Sirius meant to give Harry something very large and impressive to make up for lost years. Saying the Firebolt was ‘thirteen years worth of birthday presents’ was just a way of expressing that.
Even if Sirius remembered that first present, Harry wouldn’t have.
If Sirius did remember getting Harry a toy broom for his first birthday, he probably wouldn’t bother mentioning it in his letter. First of all, he was feeling bad that he wasn’t able to be in Harry’s life for twelve years and that Harry didn’t get to grow up knowing him. Giving Harry the Firebolt was his way of starting to make up for his absence. Harry doesn’t remember having Sirius in his life at all before, and wouldn’t remember getting the toy broomstick from him either.
“But then, perhaps it wasn’t so surprising – after all, he had only found out that Sirius was his godfather two months ago.
There was a simple reason for Sirius’ complete absence from Harry’s life until then – Sirius had been in Azkaban, the terrifying wizard gaol guarded by creatures called Dementors, sightless, soul-sucking fiends who had come to search for Sirius at Hogwarts when he had escaped.”
* - Harry Potter and the Goblet of Fire, Chapter 2 (The Scar)*
In Harry’s mind, the Firebolt would be the first present he got from Sirius. If Sirius did remember getting him a toy broom as a baby, he’d probably consider it useless and somewhat pedantic to bring it up then, especially since he couldn’t reasonably expect Harry to remember. It’d sound silly for him to write something in his letter like “Please consider it as twelve birthdays’ worth of presents from your godfather, you wouldn’t remember it, but I got you a toy broom years ago, so I really only owed you twelve presents”, despite its technical accuracy.