No. There are different signed languages around the world. The belief or assumption in hearing people, that signed language is similar around the world, is highly prevalent that Deaf people encounter in every life and in ASL 101 courses.
The term "sign language" is problematic that it's somehow perceived as a 'language' and 'modality' in one place, tangled.
Imagine -- 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.
When hearing people ask the question "Is sign language universal?", it usually refers to the language itself. 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).
"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, occasionally a few hearing persons 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?"
Wouldn't it be nice if English is universal? No.
"Learning another language is not only learning different words for the same things, but learning another way to think about things." -- Flora Lewis.
Language is culture interwined with history, tradition, belief, values, heritage, politics, etc. Language is identity.
Bilingualism or multilingualism is the way. Bilingualism, translation technology, interpreter, Esperanto, and ISL can help foreigners communicate with one another.
"A different language is a different vision of life." -- Federico Fellini.
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.
Specify a (signed) language (especially if you specify a spoken language along with it). Use the term (e.g. Auslan, French Sign Language, German Sign Language, etc) instead of "sign language".