Is a transfer function of a hole system BIBO and asymptotically stable, if the poles of the two sub systems shorten each other out?

Bibo stability is all about systems external stability which is determined by applying the external input with zero initial condition (transfer function in other words) so if you check bibo stability of G(s) ,it would be bibo stable

But

Asymptotic stability is all about systems internal stability which can be determined by applying the non zero initial condition and no external input ,and if two system as in your example are cascaded if any one of them is unstable(or both) then its modes will not decay and reaches up to infinity (in your example) after infinite time ,so G(s) would be asymptotic unstable

A system with a cancelled pole and zero may still be unstable (just not in the BIBO sense) -- it can be mathematically shown that a system with such a cancellation may still misbehave due to its response to an initial condition - this is in contrast to a stable system with all poles in the LHP, where the contribution of the initial condition to the output decays to zero. This violates asymptotic stability, but doesn't address the BIBO condition.

Furthermore, in a practical analogue realization, it is very difficult to precisely cancel a pole with a zero due to physical variation of components; your system may end up being unstable, with a pole very close to a zero in the realization.