Q
Well, I'm sure that it could be asked of another group as well, namely Muslims. They also believe they worship the God of Abraham but only in their language God is Allah. The problem is we Christians seem to have absolutely no trouble in identifying where Islam errs and so why is it that most Christians think non-christian Jews are worshipping the same God of our Lord Jesus Christ?

There can be only one way to answer this questions and it applies to all races on the face of the earth. The answer comes directly from the Son of God. This is what Jesus had to say to the kind of people that you say are worshiping the God of Israel today:

37I know that ye are Abraham's seed; but ye seek to kill me, because my word hath no place in you. 38I speak that which I have seen with my Father: and ye do that which ye have seen with your father.
39They answered and said unto him, Abraham is our father. Jesus saith unto them, If ye were Abraham's children, ye would do the works of Abraham.
40But now ye seek to kill me, a man that hath told you the truth, which I have heard of God: this did not Abraham.
41Ye do the deeds of your father. Then said they to him, We be not born of fornication; we have one Father, even God.
42Jesus said unto them, If God were your Father, ye would love me: for I proceeded forth and came from God; neither came I of myself, but he sent me.
43Why do ye not understand my speech? even because ye cannot hear my word.
44Ye are of your father the devil, and the lusts of your father ye will do. He was a murderer from the beginning, and abode not in the truth, because there is no truth in him. When he speaketh a lie, he speaketh of his own: for he is a liar, and the father of it. - John 8:37-44

The Pharisees thought they were worshiping the God of Israel and thought He was their Father. But Jesus said their father was the devil. The modern day Pharisees of modern day Israel are no different.