This is a very amusing thread. When I first saw the title, my immediate thought was Doh! Of course they wear clothes in Portugal!
But, on closer examination, I realize the OP is seeking advice on how to dress for the Camino, while still being appropriately dressed for tourism afterwards. I spent four days in Lisbon after my Camino from Lisbon to Santiago this May. Here is my take on this question.
If you made reservations for a hotel at Lisbon for the end phase of your travels, and you are starting from Lisbon, inquire with the hotel in advance, by e-mail, if they will store a locked suitcase for you (to share) for your return. For better hotels, the answer is usually yes.
This works best if you arrive at Lisbon, stay overnight before continuing to your Camino, then return to the same hotel. But, it should work even if you have to take a taxi to check your bag with the end of the route hotel.
If this works, then leave the extra/tourist clothing, shoes, accessories, etc. at Lisbon. Only carry the absolute least amount of stuff you can get away with. Of course, keep all your valuables with you at all times.
If the hotel will not hold a piece of luggage for you. And as far as I know, your sole alternative will be to ship a box of clothing from Lisbon to Santiago using the Portuguese postal system (CTT). Here are some tips.
1. At Lisbon, perhaps the most convenient CTT office (IMHO) is at the Oriente Train station. It is on street level. They will sell you boxes.
2. The CTT requires you to use a Portuguese 'return to' address. I advise using the address of the hotel you plan to use at Lisbon at the end of your Camino.
3. My choice and recommendation is to ship to the hotel you have reserved at Santiago. Check with them via e-mail, in advance, to ask if they will accept and hold a parcel for you until you arrive.
4. If #2 fails, post the box(es) to Ivar at Santiago using the instructions in this forum.
5. DO NOT ship from the Portuguese CTT to the Spanish Correos (post office using Poste Restante). I recommend against this as the Spanish post office has time limits on how long they will hold stuff at the post office for you to pick up. I do not even know if the CTT will do this.
6. Do not send anything ahead of you that is not readily replaceable. In early May, I sent a box to Ivar from Porto. It ended up lost in the Portuguese mail system for about 10 days before being spit out and mysteriously arriving at Ivar's place. I had already left Santiago to return home via Lisbon. Fortunately, I am returning to Santiago in three weeks and will retrieve this "walkabout" box then. While this has happened occasionally, it is rare. The usual time for a box to get from Porto or Lisbon to Santiago is 5 - 6 days.
Both the CTT and Spanish Correos have excellent apps for smart phones. The CTT app has an outstanding tracking system for you to track your parcel(s) on their way to Santiago. I used it for the several boxes I send to several addresses in Spain and Belgium from Portugal. I recommend getting them for you Android or iOS device.
I hope this helps.