From your exemple, it looks like your question is more about how (generalized) linear models work. I will try to give you an intuitive explanation, but others might have more accurate/complete answers.

Briefly, DESeq2 first create a model in which gene expression is explained by a formula (for instance, gene expression = base expression + treatment effect + patient effect). This is done for every gene. Then, the user can measure how significant is one of the factor of the formula (wald test) or test whether a formula is better at explaining gene expression than another (lrt test). In other words, it all start with the model, then the user does what he wants for the comparison.

If you need more details, you should read DESeq2 paper and also stat textbooks about (generalized) linear models.

This is not DESeq2 specific but rather deals with modeling in general when comparing two groups. Whether you're using DESeq2 or a t-test or some other linear model, you have to specify exactly what it is that you're trying to model.

If you assign some patients to a "treatment" group and other patients to a "control" group, you're basically looking at the difference between the two groups without caring about the individual patients within in each group. This is called an unpaired design.

If you assign some patients to a "treatment" group and other patients to a "control" group AND also specify who the patients are (e.g. patient 1 control, patient 1 treatment, patient 2 control, patient 2 treatment, patient 3 control, patient 3 treatment), this is called a "paired" design (look up how the paired t-test works). Basically, you're comparing patient 1 treatment to patient 1 control, patient 2 treatment to patient 2 control, and patient 3 treatment to patient 3 control.

Regardless, you don't get a p-value for each patient separately because you're testing the null hypothesis of whether the treatment group is the same as the control group; you only get one p-value (that is, one p-value for each gene). If your question is "does the treatment have a statistically significant effect?", it doesn't make sense to get a p-value for each individual patient.