A more interesting question is: will two persons born at the same time at the same place have the same future? Look at any twins (born within a minute or so) for example.
The obvious answer is "not necessarily". However, there are going to be some common patterns in their lives.
Planetary indications are probabilistic in nature. That probability model can be different for different people. A good astrologer studies the past of a person and makes a more intelligent assessment of the influences of various planets.
Some indications have a high likelihood and they will hold for almost all people born together (at the same place and at the same time). Some indications have a low likelihood and they may hold true only for some.
An interesting anecdote (from Prof. B.V. Raman's "Planetary Influences on Human Affairs"): In his book "Astro-Physiologic" published at Leipzig in 1928 K.E. Krafft drew up a list of 72 groups of 2 or 3 people who were born approximately the same hour, day and place, and invariably, they have died at about the same age and in a similar manner (ranging from "bronchial inflammation" to "asphyxiation" to "pulmonary tuberculosis"). The data was obtained from civil registers of the towns of Basle and Geneva.
Another interesting anecdote: I saw a program on TLC ""The Learning Channel") on TV in which they showed several twins who were separated at birth and met later in life. They found it interesting that the twins had very similar thinking styles, tastes and often same background (like taking dance lessons during certain age) despite growing separately. The theory they proposed attributed it to genes.
Higher level astrological indications that have a high probability hold for almost everyone born at that time (at that place). Lower level (fine detail) astrological indications that have a low probability hold only for a few.