If you did start your own law firm, depending on the structure of the firm you started, you could be a partner. However, if I understand your question correctly, you might be confused between the title of "partner" as a sign of seniority, and the title of "partner" as a sign of the business structure under which a lot of lawyers practise.
That is, the word "partner" does not necessarily refer to your qualifications or level of experience in the legal professional. Rather, it refers to the type of business structure in which you work and your position within that business structure. Most traditional law firms operate as "partnerships", and in a partnership two or more "partners" together own and operate the business - they are in essence the "business owners" (depending, of course, on any partnership deed or agreement).
The reason that the title "partner" is often seen as a sign of seniority has to do with the way that most partners in law firms are created. In most firms, lawyers who have worked as employed practitioners (employed by the partnership), who have shown their value to the business over a long period of time, may be determined by the partners as being such an asset to the business that they should be able to share in its ownership and management. Normally a lawyer isn't considered for partnership until they have had a number of years of legal experience and developed good relationships with clients of their "own". Hence, the title partner in most cases also reflects a title of seniority.
A large number of legal practitioners also work as sole practitioners, and some now as lawyer directors (of incorporated legal practises). In any event, if you really want to know more about this topic in detail you need to have a basic understanding of business structures - I am sure there is a wealth of introductory level material in this regard available on the net.
If, however, you started your own firm, but worked by yourself, you'd instantly be a "sole practitioner" (although arguably you don't even need to start your own firm to do it, simply holding an unrestricted practising certificate would be enough for you to work as a sole practitioner). You wouldn't be able to refer to yourself as a "partner" - this simply wouldn't make sense, as you wouldn't be working in a partnership.
I hope that was helpful!