Do you believe me when I say I see colors everywhere? If so, is that a rational belief? — Wheatley

The only rational response is none at all, to not consider the truth of your statement. Maybe you see colors in a perfectly dark enclosure, which is certainly a part of “everywhere”, but I know for a fact I cannot.

I have a widget. The widget is composed of many devices, and has produces some effect. The existence/nature of the effect, after some observation, appears to depend on its environment: when the ambient conditions of the widget change, so does the output. The output is something I can use, at least some times. Sometimes the output isn't very useful, depending on ambient conditions. But I use the output, when it is useful, to do The Stuff, which is good for me.

You claim to also do The Stuff, but it isn't clear whether you mean the same thing by "The Stuff" as I do. I ask you to describe The Stuff and your description matches mine. I ask you how you do it, and you show me your widget, and I show you mine (ooh-err!). When the ambient conditions of your widget change, you claim the response of the widget changes in the exact same way mine does.

I'm inclined to believe you have the same widget I do, and it does the same thing, but one more check. I take apart my widget and look at the arrangement of its components. You do the same. Your widget appears to have the same components in the same configuration as my widget.

What do you rationally infer? That we're doing the same Stuff with the identical widgets behaving in the same way? Or that The Stuff I do is, despite verbal agreement on the details and observable agreement in the means, still different to The Stuff you do? Are they equally likely to be true, i.e. no amount of evidence of correlation between The Stuff I do and the The Stuff you do can illuminate their similarities?

Further considerations:
1. Does it make any real difference if the response of your widget is slightly different to mine, i.e. that they are only identical in response to within some small margin of error?
2. Does it make any real difference if, instead of comparing components of our widgets (which might end their utility), we happen to know of 100,000 corpses buried with their cherished widgets plus textual descriptions on how their widgets operated and were used, and we compared our widgets to the dead people's widgets but only examined the components of the widgets of the deceased, and found 100% correlation? Would we maintain that our widgets, despite appearing the same and appearing to do the same thing, were special?