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$\DeclareMathOperator\R{R}\DeclareMathOperator\Z{Z}$No.

Indeed affine circles have been classified in the 50s by Kuiper. These are

1. the standard circle $C_1=\R/\Z$ (complete)
2. exotic circles $C_t=\R_{>0}/\langle t\rangle$ (non-complete) for $t>1$.

Any interval inside one of these circles is affinely equivalent to a bounded interval in the real line, hence to the interval $\mathopen]0,\mathclose 1[$, which is not affinely equivalent to $\R$ (e.g., because the automorphism group of the former is cyclic of order 2 when that of the latter is infinite — or because the former is non-complete and the second is complete).