There is a generalization of ultrapowers that actually gives something new, and this the socalled Boolean ultrapowers of [Mansfield, Richard, The theory of Boolean ultrapowers. Ann. Math. Logic 2 (1970/71), no. 3, 297–323] where you use ultrafilters in a Boolean algebra rather than on a set. That Boolean ultrapowers can be different from ordinary ultrapowers was shown in [Koppelberg, Bernd; Koppelberg, Sabine A Boolean ultrapower which is not an ultrapower. J. Symbolic Logic 41 (1976), no. 1, 245–249].
With real valued measures there are several problems that you can run into. First of all, if you require $\sigma$-additivity, you can usually not measure all subsets of the set $I$. So if for certain sentences $\varphi$ you want to assign a measure, namely the measure of the set of indices $i\in I$ such that $\varphi$ holds in $M_i$, you might not be able to do that in all cases. You would have to make sure that whether are or not $\varphi$ holds in $M_i$ depends on $i$ in a measurable way. While this might look like not such a big deal, you will also need such a measurability condition for formulas with parameters. And then you have to think how you are allowed to choose parameters in the individual models. That should also happen in a way that is measurable in some sense.
Another problem is that given a (measurable) set $A\subseteq I$, it can happen that neither $A$ nor its complement are of measure $1$. This is going to cause trouble when you want to prove an analogue of Łoś's theorem. The structural induction that is necessary for the theorem just breaks down in various places.
You might be able to get a reasonable structure in terms of multivalued or fuzzy logic, but I am no expert on these things and it doesn't seem to be the case that you get something interesting in the sense of classical logic.