In Birchas HaShachar we thank God for not making us gentiles by saying "shelo asni goi". Why do we use the negative formation instead of thanking God for making us Jews by saying "she-asani Yehudi" or "she-asani Yisrael"?
Magen Avraham- we want to delineate the levels of praise (see @RCW). [We need to start with some level of praise that we have mitzvos. If we would start praising Hashem that we are free or male, that doesn't mean we have mitzvos. We would have to begin by praising Hashem for "making me a Yisrael",] then you have included in that language that you are a free male and can no longer delineate. (Brackets added by me to explain MA) (O.C. 46:9)
Taz- if the brachos were said in a positive language, a person may erroneously think that gentiles or women are lower creations on the creation ladder (as in thank G-d I'm Jewish). By saying a blessing "that He did not make me...", he is saying that every category has a powerful purpose in the world and are necessary creations, but I bless Hashem for not creating me as one of the other necessary categories, since as a result I have a greater obligation in mitzvos. (O.C. 46:4)
One answer given is that we hold like Beis Shamai who said "better for man not to have been created", so we do not say a bracha for being created.
The basic Shevach, praise, of those blessings is praising Hashem for obligating us in the system of Mitzvot, commandments. We recognize there are different levels of obligation. We are appreciative that we have a greater level of obligation of commandments. A non-Jew is only obligated in seven mitzvot commanded to Noach. A slave and a woman are obligated in 613 mitzvot, but not in positive commandments that are time bound. (See Eitz Yosef quoting the Levush. He also explains the difference in obligations between women and slaves) By creating different blessings and expressing it in the negative it highlights each of the various categories of levels of obligation. Thus giving a greater praise to Hashem.
See @YDK explaining Magen Avraham
It is in the negative formulation to create parallel structure with shela asani isha and shelo asani aved. While for us parallel structure is just something our middle school English teacher ranted about in regards to Shakespeare, these prayers were written before writing was common. There were no printing presses, and the first written siddurim weren't created until the Geonic period. People were praying from memory only. Parallel structure makes things much easier to remember, which is why creating prayers like this with parallel structure was important. (You can also see many examples of parallel structure in the Mishnah for similar reasons).
Several modern scholars, beginning with the philosopher Arthur Schopenhauer, have remarked on the uncanny parallel between the wording of the Jewish blessings and an ancient Greek tradition ascribed variously to Thales, Socrates or Plato. The sage in question was allegedly in the habit of thanking God for three things: "that I was born a human and not a beast; a man and not a woman; a Greek and not a Barbarian." (Source).
It would appear then, that Chazal simply borrowed this older wording.
Incidentally, R. Aharon Lichtenstein z"l references the Greek origin of this blessing.
I have an unusual source for once. Aaron Razel (in his song We Make Ourselves) quotes R Schach as saying that Hashem makes it that we are not goyim but we are the ones to make us Jews. In other words, it is our responsibility to act as Jews, or to fulfill our potential as Jews.
I then saw here a reference to R Galinski asking that question to R Schach
Rav Shach answered that the Ribbono shel Olam grants us a neshomah, providing us with the wherewithal to soar. “But to become a Yid is up to you,” he explained. “It is up to us to develop that neshomah. You make the Yehudi.”
We all have the potential to be great and formidable. Nothing is given to us on a silver platter. We must work and strive for greatness. It doesn’t necessarily come naturally.