Is a letter from a company (specifically offering to honor a warranty for an extra 3 months) the same thing as a contract?
As you have described it, probably not, but one obviously would have to read the letter to know for sure.
A contract is a mutual agreement supported by consideration, and it is often said that the elements of a contract are offer, acceptance, and consideration. A letter that simply makes an offer (which seems to be what you have described) cannot, by itself, be a contract, although it could form the basis of a contract.
What if they send such a letter and then don't do what they said they would? Can you hold them to it?
Depends on to whom the letter was sent and what was done in response.
In other words, facts matter (which, given your post count, you ought to know), and you provided virtually none.