The Termo de Responsabilidade appears (according to Google Translate) to be roughly equivalent to the German Verpflichtungserklärung -- it is a document where the host guarantees to the government to cover the traveler's expenses and agrees that the government can hit him with the bill if you end up needing care/lodging on public expense, or if you overstay and they have expenses expelling you etc.
It makes sense that a causal friend, even one who is prepared to let you crash on his couch for a few days, will not want to risk that.
Such documents are primarily used to help applicants who can't qualify for a visa based on their own circumstances clear the hurdle.
If you're in solid economic circumstances such that you could just as well get a visa for a tourist stay in a hotel, then attaching a Termo de Responsabilidade to your application would tend to cast doubt on your application rather than help it.
In your application you will need to detail how you intend to have a place to sleep, so you need to disclose you'll be staying a the friend's house, and an informal letter from the friend will be important to help demonstrate the realism of that plan. (A copy of his ID card might be considered overkill, but probably won't hurt now that you've got it).
However, if he attempts to "take responsibilty" in that letter rather than just confirm that he intends to offer accommodation, that might backfire too (in the sense of making your application look more desperate than you need it to be, making the consular officer suspicious). Depending on exactly how the letter looks, you might want him to write a different, more casual, one for you to enclose.
If your economic circumstances can support it, it may be useful to show that you can afford the stay even if something happens so your friend cannot host you. That much may be obvious from your bank statements, or you might enclose a worst-case budget for the case that you have a falling-out with your friend and need to walk up to a hotel and pay rack rate for the rest of your trip.