Learn Portuguese (Portugal): "Obrigado" significa muito para mim.

Normally, yes! A woman says Obrigada and a man says Obrigado. However, in this sentence, you are talking about the word itself as a concept. When used as a noun like this, it defaults to the masculine form obrigado.

Is significa the standard word for saying what something means?

Yes. Significa comes from the verb significar. While it looks like the English word signify, it is the most common and natural way to express that something means something, whether you are talking about a translation, a definition, or emotional importance.

Why do we use mim here instead of me?

In Portuguese, when a pronoun comes right after a preposition like para, de, or em, you must use a special set of prepositional pronouns. For the first person, that word is mim. You only use me when the pronoun is interacting directly with a verb.

Could muito change to muita depending on who is speaking?