If isospin is conserved under strong interactions why it is represented by SU(2)?

As far as I know from my readings SU(2) is a representation group of isospin symmetry which shows deep symmetry of the strong force which conserves flavor.

Isospin symmetry is broken under weak interactions. On the other hand, Standard Model is assumed to have SU(3)xSU(2)xU(1) symmetry which here, SU(2) refers weak interaction. So if it refers weak interactions and isospin not conserved, why it is represented with SU(2) group? What is conserved rather than flavor now? I'm sure I misunderstand something please help me figure out.

(Strong) $SU(2)_F$ isospin is a global $u\leftrightarrow d$ flavor symmetry of the strong force (but not a symmetry of the EM force, and hence only an approximate symmetry).
The $SU(2)_F$ sits inside an approximate global $SU(3)_F$ flavor symmetry of the $u$, $d$ and $s$ quark, cf. e.g. this Wikipedia page.
(Strong) isospin is different from the weak isospin and the standard model gauge group $SU(3)_C\times SU(2) \times U(1)$.