One can think of the expression “être à la hauteur de” to express the idea of the verb to match. But I am not sure that an universal expression could be found to translate it and to give back the same meaning in all the context. In some cases one can use the verbs “égaler”, “être comparable à”, “équivaloir” (this one can be tricky to conjugate) etc.

• Depending of the context, correspondre, égaler, or être à la hauteur/au niveau can be good translations:

Correspondre:

Son habillement correspond à ses revenus.

or

Votre offre correspond tout à fait à mes besoins.
(Your offers does perfectly match my needs.)

Égaler:

Il est difficile de t’égaler au tennis.

Être à la hauteur:

Il travaille dur, mais son salaire n’est pas à la hauteur de son implication.
(He is a hard worker, but his salary doesn’t match his dedication.)

• This is not a comprehensive list as this is really context-dependent. To my knowledge, no French verb is as generic as to match.

• And quite frequently I can hear the anglicism matcher, in France. You might be led to hear that, especially at work:

*Ton implication et mes attentes matchent, c’est bien !
(You dedication matches my expectations. That’s good!)

This is actually bad French, so this should be discouraged.

If by

• Her dress matches her purse;

you are speaking of color and style (and so purse is not here as an euphemism for wealth), one would say

Sa robe est assortie à son sac.

To find a french term which match a desired meaning in the large set of meanings of to match, I’ll recommend again this site (sadly, I’m unable to give a direct URL to the map it generates) which gives 60 or 70 French terms and maps them along about the same number of English words more precise than match.