How do various frameworks account for situations when multiple cases can be assigned?

Case assignment is a big problem for formal frameworks that postulate some sort of deep agentive case-like structures like theta roles in GB because whenever the morphology clashes with the deep structure. It's not too difficult to deal with it as part of the morphology module but cases like this have to be dealt with as an exception.

On the other hand, it is not a problem in the least for functionalist or construction-based approaches. They simply look at it as a pattern fulfilling a function. Of course, your example

My mother and me went to the market.

Also has the hypercorrection counterexample:

Jane went to the market with my mother and I.

which I suspect would also be treated by many as grammatical (in non-prescriptivist contexts). In both of these cases, we're dealing with a coordinate construction where 'me' and 'I' are interchangeable as long as there is some cognitive shielding to the infelicitous interpretation. Note, that sentences

*Me went to the market.

*She went to the market with I.

are both not acceptable. This means that the idea of the case function-form pairing itself has not been lost in English. Something just happens in the coordinate constructions with 'me'. This is similar to the loss of 'whom' in English where 'Whom did your see.' is outside most native speaker intuitions whereas 'Her I saw.' is not.

English does not have a nominal case morphology so you can dismiss these examples as easy to file away surface exceptions. But you can find examples even in languages where the morphological case is unstable in even in languages where it is unavoidable. My favorite example is the Czech verb frame 'teach something to someone' which in modern Czech has both the direct and indirect object assigned the accusative case. That's because the direct agentive structure makes sense when either of the objects is on its own.

Jane teaches math (acc).

Jane teaches John (acc).

And therefore,

Jane teaches John (acc) math (acc).

However, in formal (now slightly archaic) Czech, the frame is

Jane teaches John (acc) math (dat). [Note: The frame is similar to the English 'introduce X to Y']

You can see that in this case, you can't just shove this into some surface morphological exceptions on top of a stable underlying agentive structure. There are three possible case assignments 'acc acc', 'acc dat', 'dat acc' which both map onto a plausible semantic structure but only two are acceptable in Czech morphologically.

The solution in a formal system would have to be to move this problem into the lexicon which is why it is much easier to deal with it constructionally where the division between lexicon and syntax is not as sharp and morphology is not just an afterthought.

Note: I'm not sure how this would work under Minimalism but since that is essentially a unification approach, it may not be that hard.