It's completely reasonable to use a paired t-test when the two samples are not the same individuals, as long as they are meaningfully paired in some way. Conducting an independent samples t-test and a paired t-test asks very different questions, though.
An example, to illustrate
Let's say you want to test whether teenagers differ from their parents in political orientation, assuming a simplified left-right continuous political scale where 0 means far right and 10 means far left. In general, parents and their children will probably be relatively close to each other on the scale (i.e. conservative parents will be more likely to have conservative kids, and liberal parents will be more likely to have liberal kids). But perhaps teens tend to be more left-leaning than their parents, so the child of a conservative parent may be a little less conservative, and the child of a liberal parent may be even a little more liberal.
If you conduct an independent samples t-test, it will answer the question "Do parents, overall, differ in political orientation from teens, overall?" It will test whether the mean political orientation in parents is different from the mean political orientation in teens. A paired t-test will answer the question "Do teens differ in political orientation from their parents?" It will test whether the mean difference in political orientation for all of the parent-teen pairs is different from zero.
It's not clear from your description whether you want to look for overall differences between the means of the two samples, or whether you want to know about the difference scores for each matched pair. It is completely reasonable to conduct either the independent or paired analysis --- you should select whichever one will best answer your research question.
Another option which might feel more intuitive for you, depending on how this "matching" process worked, is an ANCOVA. You can control for the matching variable (height, weight, whatever), and look for differences between the groups after partialing out that variable.