Learn Portuguese (Portugal): O gato gosta daquele tapetinho fofinho.

To add the -inho suffix to a word ending in a vowel, you must drop that final vowel first. So, tapete (rug) drops the -e and adds -inho, becoming tapetinho (little rug).

Why did we add -inho to the adjective fofo to make it fofinho?

While fofo already means fluffy or cute, adding the diminutive suffix to an adjective intensifies it or gives it an extra tone of affection. Using it here makes the rug sound extra cozy and appealing to the cat.

Do I always have to use the -inho ending on the adjective if the noun has it?

No, it is not a strict grammar rule. You could easily just say tapetinho fofo. However, Portuguese speakers naturally love to match the diminutives on both the noun and the adjective to really emphasize the cute, small, or cozy nature of the object.

Why is the word daquele used here instead of just aquele (that)?

The verb gostar (to like) always requires the preposition de (of) immediately after it. When de meets the demonstrative word aquele (that), they merge into a single word: daquele. So in Portuguese, you literally say that the cat likes of that little rug.