Reading the background text, I noted the following:
Abram's clan of 318 men had accomplished a great victory! They had gone up against a formidable army of four victorious kings, and won!I have not studied Melchizedeck, but it follows that God had revealed Himself to Melchizedeck and called him into a royal preisthood much as He had revealed Himself to Abram and called him into the establishing of a people, sanctified and holy to God, El Elyon - thus the Kingdom of God.I also note the place - Salem, which became known as Jerusalem. King David went to great lengths to free the city and established his kingdom there. I beleive (this is from past reading of the scripture but I have no biblical reference to support my statement so this must be given as my thinking at this point - I do intend to search it out for my own edification in my journey of discovery) that God's plan from creation was to establish His kingdom through Jesus Christ here on earth and specifially pinpointed it to take place at Jerusalem, Peace, and from there it would fill the whole Creation. Furthermore, God claimed the throne of David as His own and placed His name there forever. When Christ returns, Jerusalem will be the seat of His government.The victorious return was greeted by both the carnal and the Godly. Abram had to make a choice. Riches from the spoil, or the wealth of God's blessing, uttered by Melchizadek. We assume from verse 22 that Abram had sought God prior to going out to battle, making it a righteous war in that he did not spill blood for the sake of personal gain. When faced with the choice, he chose to keep his vow. This had been a test. The assurance that he would be protected by El Elyon, and that his reward would be VERY great follows in Chapter 15:1. Melchizedek's prayer had been more than a blessing, it had been prophetic and intercessory - a prayer for the establishment of God's kingdom on earth, beginning with Abram's obedience and righteousness. Today, we are instructed to pray for the peace of Jerusalem - the return of Christ.The communion, unity, sharing between Abram and Melchizedek included God. El Elyon was central to every part of their lives. El Elyon had revealed Himself to them each as a personal, living, powerful, trustworthy, loving and Sovereign God. Melchizdedk brought out bread and wine: Just as in later times when Moses set up the Tabernacle and God established the sacraficial laws, the show bread was renewed daily and the wine (blood - life) was poured out for the life of the sanctified people of God. This also was prophetic of Christ.Finally, the tithe. It was a worship and thank offering to God. God sustained Melchizedek and added to his sustenance through Abram's worship of the Most High God. When we do not give our tithe freely and joyfully, it becomes a payment, a bribe or sorts, rather than a recognition of God as our provider; and we must share His provision with others if it is to be increased to us as individuals.
So, to answer the question, "What did Abraham and Melchizedek seem to share in common about God Most High?" -
They shared the common belief that El Elyon is the One true God, Sovereign, and working to establish the redemption of His people, reconciling His Godly ones - any and ALL who respond to His call - to Himself.