Learn Turkish: O yalnız kedi benim tek sırdaşım. - That lonely cat is my only confidant.

The base word is sır (secret). The -daş suffix (which can also be -deş, -taş, or -teş) implies sharing, fellowship, or commonality. By attaching it, we get sırdaş (someone who shares secrets, i.e., a confidant). Finally, the first-person possessive -ım makes it "my confidant".

Both yalnız and tek can sometimes translate to "only". How do their meanings differ in this sentence?

Yalnız primarily means "lonely" or "alone" as an adjective, which is how it is used to describe the cat here. Tek strictly means "sole" or "only one", indicating that the speaker has no other confidant. While yalnız can occasionally mean "only" (as an adverb meaning "just"), tek is the natural choice for "the only [noun]".

The word sır often doubles its final consonant when taking a suffix (like sırrı). Why doesn't it double in sırdaş?

Arabic loanwords ending in a single consonant, like sır or hak, only double their final consonant when a suffix starting with a vowel is added. Because the fellowship suffix -daş starts with a consonant, the root stays exactly as it is: sırdaş.

There is no verb in this sentence. How is the "is" understood?