Your question is more to do with legalities of the will than taxes.
The $13K he refers to is a gift tax
exemption. He is telling you that to avoid GIFT tax on your brother, he can only gift you $13K per year. This is only half-true.
He can gift you an amount up to a million dollars in one year, but he would need to file a gift tax return to claim part of his lifetime exemption. If your brother's entire estate is less than a million dollars, this will not affect him now or later in life. If we're talking about a $100K, again, he can file the gift tax return and he will be exempt from the gift tax.
Recipients never pay gift tax.
The first question I would ask an attorney, though, is whether he needs to split the inheritance of the house in the first place, which it sounds like he must. There is no gift or estate tax
on this transaction
. It needs to be titled properly.
Your uncle probably made the mistake of willing the house to both of you, meaning neither can sell it or do anything without the other's agreement. His intent was to give you half the value each, not to take a chain saw and split the house in two halves.
As such, he has made a mess, I am sure, and quite unintended.