It’s not a secret that the sine, tangens and the x-square function are almost equal around the positive side from zero, but in the end, they only have one common point. Do two functions exist, that share an entire interval, but are different.
[With elementary function meant is anything using the basic operation, and defined in one term.]
Not possible for "analytic" functions (that is, functions that are sums of their power series). If sums of two power series agree on an interval, then the two series are identical.
It can happen for non-analytic functions, even $C^\infty$ functions. The best-known is $$ f(x) = \begin{cases}e^{-x^2}, &x>0,\\0,\quad&\text{otherwise}\end{cases} $$ Here the function $f$ and the function $0$ agree on $(-\infty,0]$.