Hey J,
Birch has been used for plenty of axe handles, this because it is a bit softer than hickory and elm which will relieve some stress given to the joints of the user.
However it being softer, and a little lower in the tensile strength, so in a really heavy or long handled axe like a splitting maul or a full size felling Axe I would advice Birch however it is still a possibility, You will however need to replace a birch handle a good bit sooner then you would need to with a Hickory, ash, or Elm one.
Just so you know and Here is a little story to bring it all into perspective:
As we all know White oak has a really high amount of strength and force it can negate, and If I can recollect it correctly English White Oak( Quercus Robur) could withstand about 12000kg in the radial direction ( per Well I cant remember the standard sizes used to calculate that XD) and Birch came up to about 8600kg with hickory around the 9800, so it has a bit less tensile strength indeed but you wouldnt have to worry about breaking anytime soon.
Hope that helps bringing it into perspective ^^
And stick with the tight grained birch, and if you have British native birch be carefull, as the quality is allot lower than the Norse/Russian Birch on which the tables I described above are take from.
Yours sincerely Ruud
PS grain alignment does become a really important thing with birch so keep that in check!