You are right that it is not a gift like a book or a piece of jewelry. It is a gift, or "thing", of infinitely greater value, freedom. When a master releases his slave from bondage, do you think the slave perceives that a gift has been given to him? I certainly think so. The former slave will absolutely carry his freedom with him for the rest of his life. Likewise, the recipient of God's saving grace will carry his freedom, salvation, the Holy Spirit, and more with him for the rest of his life.
Rationalizations, FK. Gifts are things we give for Christmas, birthdays, Valentine's Day, etc. Freedom is freedom. If someone is freed, he is either pardoned, or released, or "made free." Mercy is just what the word means — mercy; being spared.
Reformers believe that only those predestined to receive mercy will receive mercy, so it is no gift but an 'executive' (no pun intended!) decision before all ages before those receiving mercy were even created (and 'enslaved').