I have looked at the photos of the two plants and I do not think they are the same cultivar. Your neighbour's plant seems to have different shaped petals and sepals and their colour seems different from your plant. It is very difficult to be certain about flower colours as cameras and lighting can change the appearance. Looking at the two photos I see a difference in the green colour of the leaves so that makes it more difficult to be confident that the flower colours of the two plants really are different.
However, I am going to conclude that they are different cultivars. Yours is 'Stella de Oro' and your neighbours is some cultivar that is related to Stella.
I am going to assume that most pods on both plants are from cross-pollinations. That tends to be the case under natural conditions. We often assume that because Stella de Oro is self-compatible that most pods are from self-pollinations but simply looking at its seedlings can give us a clue about whether that is the case. For example, all seedlings of Stella that have red or other coloured eyes will have been from cross-pollinations.
We also tend to assume that because Stella is self-fertile that it will be compatible with all other daylily cultivars. That is probably not correct and in daylilies there is no reason why that should be the case. Stout looked at self and cross compatibilities in a set of sibling daylilies and there were four of 12 siblings that were self-fertile. However, even thought they were self-fertile all four were incompatible with some other siblings.
Theory 1. Pods on both daylilies are from cross-pollinations. Your neighbour's plant is compatible with Stella pollen. Your plant ('Stella de Oro') is not compatible with the pollen from your neighbour's plant. Some self-pollinations (by no means all) of your plant should produce mature pods and seeds.
Theory 2. Some self-pollinations are happening. Your neighbour's plant is self-compatible and some of its pods are from self-pollinations. The pods on your plant are nearly all from pollen from your neighbour's plant and that is not compatible pollen.
A proper test of any possible theory would need many flowers on your plant and pollen from your plant and your neighbour's plant.
In any case, seeds from self-pollinations or from pollen of your neighbour's plant would probably only produce yellow small-flowered seedlings; probably not what you were hoping for. You could use any seeds you produce from such crosses in learning how to grow daylilies from seed.
