The gender of a noun is just a noun class. But because Germanic languages lie more toward the analytic end of the morphological typology continuum (whereas Romance languages lie more toward the synthetical end) the information is latent - or rather, the task of conveying that information is left to other words (the articles).

Just imagine if someone studied Portuguese but learned vocabulary like this, never bothering with the ending vowel:

  'gat-'

  'cas-'

  'bolach-'

Similarly, 'die' should be considered an inherent part of 'Frau'. So don't learn just 'Frau', learn 'die Frau'. The article 'Die' is just as "random" as '-o' or '-a' is in Portuguese. (I'll skip the part where you can have a form of the word in both classes: gata/gato.) People like to try and find "rules" they can remember instead, but it's a pointless endeavor. Language is a Calvinball game.

To make a weird tech analogy: Romance nouns are like laptops, with a touchpad built in. Germanic nouns are like desktops, you have to remember to carry a mouse* along.

* Die Maus

From a foreign learners perspective, it is easier to just learn the article together with the noun.

But there are rules for 2/3 of cases. https://sprachekulturkommunikation.com/genus-der-substantive...

You can classify by suffix.

* -ung, -heit, -keit -> feminin, e.g. die Schönheit

* -ling -> masculin, e.g. der Flüchtling

* -chen, -lein -> neutrum, e.g das Mädchen

You can classify by category. Every alcoholic drink is masculin, except for beer.

You can classify by phonetic spelling. That is probably the closest you have to Portugese.