You impression is wrong, sorry. In (modern) Italian, the participle in a verbal form constructed with the auxiliary essere agrees with the subject.
Renzo si è lavato le mani e si è messo la mascherina, poi è entrato nel negozio.
Lucia si è lavata le mani e si è messa la mascherina, poi è entrata nel negozio.
You could find in old books agreement with the direct object for reflexive verbs, something like
Renzo si è lavate le mani e si è messa la mascherina, poi è entrato nel negozio.
but this is not the current majority usage. Some people might use it, in order to use parlare ricercato (refined speech); I believe it's still an alive form in Tuscany. See Concordanza in the Enciclopedia Treccani. The case of the composite forms of the verb essere itself is a bit different, but it's not this case.
In French the participle agrees neither with the subject (elle) nor with the direct object (les mains): otherwise it would be lavée (which is not wrong, as far as I know) or lavées. Different rules.
Note that when the verb is constructed with the auxiliary avere (for example in the active passato prossimo) the participle usually doesn't agree with the subject. Actually the rule of agreement in this case is quite complex, see Past participle and changing endings with auxiliary verb "avere"