Is sign language universal?

No.

How about this -- Is speech language universal?

Yes *tongue-in-twist*. Well, if you're talking about the modality that hearing people use speech more than signing. If you're referring to language, the answer is obviously no.

Of course, when hearing people ask the question "Is sign language universal?", they mean the language itself, not modality. For the sake of clarity, let's refer "signing" and "speech" to as modalities and "signed language" and "spoken language" as languages.

By modality, there are three or four forms: writing (for virtually everyone), vocal-auditory speaking (used by hearing), visual-spatial speaking (used by Deaf), and tactile-spatial speaking (used by Deaf-blind).

It's not unusual that many (actually ultra-very common) hearing people assume that either sign(ed) language is universal or wonder whether it is. The term "sign language" is somehow vaguely referred to both language and modality that become tangled.

Deconstruction

"I know little sign language." "I learned some sign language." "Growing up, I've been learning English and French. Now I'm learning sign language."

"Which language?" I (as the ASL instructor) asked to help deconstruct the student's mindset. Previously I taught ASL students, "I'm learning ASL."

The student replied, "sign language," I nodded, "which language?" The student, looking puzzled, repeated it. Irrestistibly, I repeated, "which language?" She/he, looking somehow more confused or sometimes frustrated, "sign language?". Generously, I repeated, "which sign language?" If a student doesn't get it, usually their classmate helps reply, "ASL." Often, the student's little light bulbs flashed in their eyes.

So, yes, just like spoken languages, there are different signed languages around the world. It's not universal.

Most of the time, if not virtually all the time, hearing people accept this fact and move on. Fantastic.

But, worse, a few hearing people would find this fact a sore disappointment. "Would it be nice if sign language is universal? So that everyone around the world can communicate with each other. Wouldn't it be fabulous?"

Isn't that naivety?
Reaction meme for facepalm

1) Again, which [signed] language? Don't say "English/Spanish/French/etc. and sign language". This implies a hierarchial structure. 2) Wouldn't it be nice if English is universal? No.

Language is culture interwined with history, tradition, belief, values, heritage, politics, etc. Language is identity.

Language as Identity

Bilingualism or multilingualism is the answer. But, monolingualism around the whole globe? Not happening.

Bilingualism, translation technology, interpreter, Esperanto, and ISL can help foreigners communicate with one another.

We, Deaf ASL-speaking people, value signed languages of their own countries. Multiple languages and cultures around world along with a universal respect are a beautiful experience.

Takeaway

Make a habit of specifying a (signed) language. Use the term (e.g. Auslan, French Sign Language, German Sign Language, etc) instead of "sign language".

Use a specific language for one of the signed languages too if you use a specific spoken language.