The Standard Model of particle physics can be constructed by specifying its gauge group $G$ and the representations of the fields (plus some extra information: Lorentz invariance, values of the coupling constants, etc.). Using this model we can predict many experimental facts, such as the cross sections measured in collider or confinement/asymptotic freedom of QCD.
However, we could've started by just giving the Lie algebra $\mathfrak{g}$ of $G$, the representations of the fields under $\mathfrak{g}$, and so on. For the purpose of making many of these predictions, it suffices to work at the infinitesimal level.
On the other hand, the actual group $G$ is important in gauge theory, as we can see, for example, when dealing with instantons. So it actually matters, for some applications to be able differentiate between different gauge groups even if they have the same Lie algebra.
Is there any experimental evidence or theoretical reason for $G=SU(3)\times SU(2)\times U(1)$ instead of some other group with algebra $\mathfrak{su}(3)\oplus \mathfrak{su}(2)\oplus \mathbb{R}$?
This Phys.SE question is clearly related, but it's only asking about a concrete modification of $SU(3)\times SU(2) \times U(1)$ to take into account a relation between the representations of the matter fields, not about exploring other possibilities for the global structure of the group.
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