Only in a way. Your analogy is actually showing something called "modalism" or the idea that one (unknowable, and hidden) person of God plays 3 roles. It was a heresy condemned in the 3rd and 4th Centuries. It's related in a way to unitarianism as it assumes there's one monad God, which we cannot really know, as he just uses 3 different masks, or hats, like your business owner--and we only see him behind those masks.
This is kind of the opposite to Mormonism, which is at least tri-theism, but really polytheism. Another interesting thing about Mormonism is most forms tend to believe matter and the universe is eternal....with an endless sucession of gods begatting gods going back ad infinitum. Mormons cannot confess that God made all that is out of nothing.
The classic formula attempting to know a little of the nature the infinite God says God is one in essence and three in persons.
Another way to put it is how it was described since at least Augustine (AD 430). The Father is God, the Father is not the Son or the Holy Spirit. The Son is God, the Son is not the Father or the Holy Spirit. The Holy Spirit is God, the Holy Spirit is not the Father or the Son.
Do I understand it? No way. Is is surprising that a finite creature cannot grasp the concept of 3 persons who are one being? No....as its quite logical that the infinite eternal only high God's inner nature would be a serious mystery to us.
There are all kinds of analogies, but every one fails at one point or another. 3 parts of an apple (skin, flesh, seeds) yet every part is apple. 3 leaves of a clover, but just one clover leaf. 3 dimensions.... 3 states of water (liquid, vapor, ice). The 3 parts of time (past, present, future).3 (major) parts of an Atom. 3 parts of ethics (the standard, the situation at hand, the individual involved), etc, etc. It seems the harder one looks, the more triads there are regarding concepts in the world. (but of course "triad" is just a word picture of abstract reality).
The problem with the idea of God playing different roles, as I hinted above, is that we cannot know someone who's always behind a mask. However that's not how scripture speaks of God. It says we can know Jesus Christ personally, and to do so, knows God--and simultaneously comes to know the Father--through the power of the Holy Spirit.
The Father sent Jesus. Jesus came and obeyed--even to the cross and resurrection. The Holy Spirit gives us new hearts to receive this (and understand, some). This formula shows what is called the "economic" Trinity--different roles yes, but unlike your business owner, 3 distinct persons too.
A pastor I greatly trust advises that when speaking of the Trinity its not wise to rely on analogies--since none can capture trinitarianism just right, but to just take what scripture says. It refers to Jesus as God, and yet Jesus prays to His Father, as God, and it speaks of the Holy Spirit as God. And throughout, Old Testament and New, it also says there is only One God (the famous sh'mai, of the Jews). The one God issue....in and beyond all the universe, is the biggest stumbling block for Mormons.
And no, followers of Joseph Smith or Brigham Young, Jesus was not the spirit-brother to the Devil, or to you and me. Angels, fallen or not, are a different species entirely than human beings. God was father only to Adam, and Adam disowned that. He only becomes Father to us again, when we trust and thereby become unified with Jesus, called the 2nd Adam in the NT book of Hebrews.
Outstanding response. I agree with everything you said.
Go back to post #83 which was made in response to a question concerning the Trinity and you will find a Biblically correct reply.
Our exchange continues through postings #104, #123 and #148. You will see where I used an admittedly weak answer, but the best available to at least begin to introduce an individual to the Bible.
I have employed this approach with mixed results (usually success if our conversation continues for some period of time) in the past when discussing the Christ revealed in the scripture with persons who have a incorrect and/or incomplete picture of Him through a false religion and thereby find the Trinity a stumbling point. The two most common examples of this are Muslims and Mormons.
I admit my analogy is imperfect, and that none is, but it is better than none, and you have to start somewhere.
Incidentally, you have presented an especially well reasoned and erudite response by Free Republic standards, which generally greatly exceeds those on most other political forums. I congratulate you and would love to learn more about your background.