Do Christians receive the same inheritance as Messianic Jews? I know that in Christ there is complete redemption for both Jew and Gentile but it seems as though God has made specific promises to the house of Israel and that God throughout the bible shows special affinity toward Israel. 

The promise of Israel was that they would rule all the nations from Jerusalem and that all the material blessings of God will flow from Israel to the other nations. 

God also refers to Israel as his firstborn among the nations and according to the royal pattern God has set, the first born always receives the largest inheritance among his siblings.

So in patristic perspective, do the Jews in Christ receive a particular inheritance while Gentiles also receive a great yet definitively different inheritance?

The Jews in Christ will inherit the land of Israel. Will Gentiles rule in Israel with Jews or will Gentiles inhabit / rule their own respective nations in Christ, i.e Russia, China, Central African Republic, Japan, Germany, India, etc.? What is the protestant view on this?
Most Christians will likely agree on the basic principle that the Children of Israel, down to the present day have a special relationship with God, but that at this stage, we don't know what that relationship is, exactly. Jesus' parable of the workers in the vineyard (cf. Matthew 20:1 ff.) might apply here, with the Children of Israel representing those hired earlier in the day, and Gentile Christians representing those hired at the eleventh hour. On the other hand Paul taught the Galatians (3:28-29) that there is "neither Jew nor Greek, slave nor free, male nor female, for you are all one in Christ Jesus. And if you're Christ's, then you are Abraham's offspring according to the Promise." [RSV]

Besides, Jesus was a Jew. Christians believe nearly universally that we are part of the Body of Christ, which makes us Jews, too.

By the way, it should be noted that depending upon their exact beliefs, most Messianic Jews would be considered Christians.