ANSWER: The first thing we near to clear up is that Jesus is NOT the Father. Though of the same character and mind, Christ is a separate Being from the Father as many Bible verses attest to (see John 5:18 - 19, 5:37, 14:26, etc.). He is God, the Son, our Lord and Savior.
Your question concerning why people are baptized using more than the name of Jesus is a good one. It has troubled quite a few people. No where else in the scriptures is the command given to be baptized in the "name OF the Father, and OF the Son and OF the Holy Spirit" except in Matthew 28:19. Compare this with below.
"And he commanded them to be baptized in the name of the Lord. Then they asked him to stay a few days." (Acts 10:48)
"When they heard this, they were baptized in the name of the Lord Jesus." (Acts 19:5, see also Acts 8:15 - 16, Acts 2:38)
There are many passages in the New Testament about doing things "IN THE NAME of the Lord" or Jesus (like casting out demons). There is no Biblical instruction to do anything "IN THE NAME of the Father" or the Holy Spirit. (Colossians 3:17)
If you do a word search, using Strong's Concordance or a computer program, using the phrase "in the name of" you'll find no reference to actions taken by the Father or the Holy Spirit.
I also checked the original Greek scripture for Matthew 28:19 and there is no other way to translate the actual Greek words of that passage. Nor are there any footnotes in any of the translations I can find designating the passage as "suspicious" or "questionable." See Mark 16:9 - 20 and 1John 5:7 - 8 which some translators say were either not there in the oldest manuscripts (the former) or slightly different in original manuscripts (the latter).
Another fact to consider is that all our English language Bibles are translations from the Greek and Hebrew (or Aramaic) languages and that syntax (the way words are put together to form phrases, clauses or sentences) is completely different in English, Greek and Hebrew (and many other languages). So there may be punctuation, or phrasing differences in the syntax which would change the way this passage reads.
However, I have no Biblical or secular justification for saying that the above passage contains any error. Should we use this single passage to justify baptizing a believer (or ourselves being baptized) using that phrase? Each of us will have to make that decision on their own. God is understanding and forgiving. Certainly we would not be blaspheming to use the phrase.
The baptism of a believer should be done "in the name of Jesus," corresponding to Colossians 3:17 and all the other passages which use this term, including the Bible verse found in Acts 4:8 - 12.
In summary, it is never a good idea to base an idea or a doctrine on a single passage of scripture. The Bible is "multiple - redundant", i.e., all the things that are vital to our salvation and our knowledge of God are repeated throughout the Old and the New Testament canon so often that if one studies and compares scriptures, we may find that a single passage is misunderstood, or mistranslated or that it is not important to our salvation. So it is with this passage.
The important thing is to believe on Jesus, repent of our sins, confess His name before men and to be baptized for the remission of our sins and to come up from that symbolic grave with a determination to change our sinful ways, to learn the ways of God and to live by His commandments to the best of our ability, with prayers to Him to help us live that way and for greater understanding of what He would have us do.