How do we know we aren't worshipping the same God? How does a Catholic know that they are not worshiping the same God as someone who practices Islam or Judaism?  Is there anything that concretely points to them being different past personal revelation?  Has the Church expressed in a doctrine the differences between their God and the others?
The Catholic ought to start with

NOSTRA AETATE DECLARATION ON THE RELATION OF THE CHURCH TO NON-CHRISTIAN RELIGIONS PROCLAIMED BY HIS HOLINESS POPE PAUL VI ON OCTOBER 28, 1965. In it Pope Paul VI states


  3. The Church regards with esteem also the Moslems. They adore the one God, living and subsisting in Himself; merciful and all-
  powerful, the Creator of heaven and earth,5 who has
  spoken to men; they take pains to submit wholeheartedly to even His
  inscrutable decrees, just as Abraham, with whom the faith of Islam
  takes pleasure in linking itself, submitted to God. Though they do not
  acknowledge Jesus as God, they revere Him as a prophet. They also
  honor Mary, His virgin Mother; at times they even call on her with
  devotion. In addition, they await the day of judgment when God will
  render their deserts to all those who have been raised up from the
  dead. Finally, they value the moral life and worship God especially
  through prayer, almsgiving and fasting.
  
  5. Cf St. Gregory VII, letter XXI to Anzir (Nacir), King
  of Mauritania (Pl. 148, col. 450f.)
  
  4. As the sacred synod searches into the mystery of the Church, it remembers the bond that spiritually ties the people of the New
  Covenant to Abraham's stock.
  
  Thus the Church of Christ acknowledges that, according to God's saving
  design, the beginnings of her faith and her election are found already
  among the Patriarchs, Moses and the prophets. She professes that all
  who believe in Christ-Abraham's sons according to faith 6
  -are included in the same Patriarch's call, and likewise that the salvation of the Church is mysteriously foreshadowed by the chosen
  people's exodus from the land of bondage. The Church, therefore,
  cannot forget that she received the revelation of the Old Testament
  through the people with whom God in His inexpressible mercy concluded
  the Ancient Covenant. Nor can she forget that she draws sustenance
  from the root of that well-cultivated olive tree onto which have been
  grafted the wild shoots, the Gentiles.7 Indeed, the Church
  believes that by His cross Christ, Our Peace, reconciled Jews and
  Gentiles. making both one in Himself.8
  
  The Church keeps ever in mind the words of the Apostle about his
  kinsmen: "theirs is the sonship and the glory and the covenants and
  the law and the worship and the promises; theirs are the fathers and
  from them is the Christ according to the flesh" (Rom. 9:4-5), the Son
  of the Virgin Mary. She also recalls that the Apostles, the Church's
  main-stay and pillars, as well as most of the early disciples who
  proclaimed Christ's Gospel to the world, sprang from the Jewish
  people.
  
  6. Cf. Gal. 3:7
  7. Cf. Rom. 11:17-24
  8. Cf. Eph. 2:14-16


The God whom the Church, the new People of God, believes in, they owe it to the revelation of the Old Testament which the Church received from the Jews, God's own chosen people.

The Catholic may then wish to move on to "Muhammad?" and "Judaism?" sections in Crossing the Threshold of Hope by Pope St. John Paul II [the Great].

The Catholic should then conclude that from these papal teachings, the Catholic Church acknowledges that Jews and Muslims worship the same God. (OP: Even though Jews have a incomplete knowledge of him and Muslims have an incomplete and distorted knowledge of him).



The context in which these declarations should be viewed


Worshiping the same God does not imply that the various worshipers are worshiping God as he ought to be worshiped. cf. Jesus and the Woman of Samaria.
Among those who worship God, some worship is acceptable to him and some is not. cf. Cain &amp; Abel and Peter and Cornelius. In the case of Cornelius, even though God accepted him, God via Peter brought him to the fullness of the Faith. 
(OP: This declaration is not an endorsement of the Jews' and Muslims' worship, much less of their religion).