Jews and Muslims (along with Mormons, Unitarians, Jehovah's Witnesses etc.) don't worship a completely different god (like say Zeus) but they worship an utterly false version of God by denying the Trinity and/or the fact that Jesus is the Son of God and moreover Saviour.
I believe Christians worship the God of Abraham, Isaac, and Jacob. We are grafted in. Jesus was a Jew. We make a mistake if we ignore the roots of our faith. The issue here is what Jews believe about Jesus.
As for Muslims and others, they worship a the god of this world in a variety of forms and ways. They do use the same terms but, as Squire put it, they work from a different dictionary.
John 8:19They said to him therefore, “Where is your Father?” Jesus answered, “You know neither me nor my Father. If you knew me, you would know my Father also.”
1 John 2:23 No one who denies the Son has the Father; whoever acknowledges the Son has the Father also.
John 16:3 They will do such things because they have not known the Father or me.
You can probably make an argument from the 1 John reference. It would indicate that they are not saved. If the topic of thread were different then we would be discussing whether or not God is done with the Jews and the nature of the Covenants He made. However, this thread is specifically about whether or not they are worshipping, praying to, etc the same God we do as Christians. That being the case, I would still have to say...yes, they are still trying to commune with the God of Abraham, Isaac, and Jacob and He is our God. If you want to argue about how effective they are in doing that, based on what they believe about Christ, that is a matter for another thread.
As for the verses you chose from the Gospel of John the context is important to note. Don't confuse statements made to specific people or crowds with blanket statements to the Jews as a whole.
Jews do not believe in or worship the Triune God.
The Triune God is the true God.
The Father of of Jesus is one person of the Triune God.
Without the other two persons God the Son and God the Holy Spirit, you do not have the Triune God.
Read carefully your statement.
You could easily be misinterpreted as saying Jesus is not the true God.
jesus Himself verified that the jews call His Father their God, and that is correct, but that unless they also acknowledge Him as the Son, the Messiah, they will not be in Covenant relationship, for now they need to depend upon relationship to the father thru christ, just as gentiles do!
they know of the real God, but when they denu jesus, they are NOT in
covenant relationship with God!
There probably shouldn't be any ifs or howevers surrounding this question. It's about as absolute as you can get.
They are trying to communicate with as errant a version of God as do the Mormons and Muslims. The fact that they still go to hell testifies that they are not worshiping the same God as Christians but rather their version of who they want God to be.
The Good News was given to the Jews. They rejected it in favor of their own tradition. That's idolatry.
Actually, the majority rejected jesus, as per the predestined plan of God to graft in gentiles to salvation, but
the Jews still have the OT that IS from God, and do pray to the true God, but are NOT in a Covenant relationship with Him apart from believing in Jesus as their Messiah!
Jews know of the true God, and have inspired OT texts from Him, but NOT in a saving relationship unless faith in Yeshua!