I understand that the alphabet for the English language is not strictly English as languages such as French, Dutch and many more use the same alphabet, with few additions in other languages. Is Roman alphabet the common term to refer to this set of symbols for writing texts?
The term for the name of the script is Latin, at least according to Unicode.
Contrast those with non-Latin letters, like these:
Any code point with the Unicode character property
Bringhurst in Elements of Typographic Style talks about how the Latin alphabet is not the ASCII alphabet, but contains in fact hundreds more glyphs, or sorts as a typesetter might say. Most of these we only use in English when writing an unassimilated loan words, or someone’s name, if at all.
And it gets weirder, because as more world languages have adopted the Latin alphabet, they often have reason to add new letters to it for their own needs.
"Roman" is a type of lettering based on the letter-forms adopted in Ancient (Imperial) Rome:
[Image credit: Me]
The name is commonly used for the writing of a number of languages, as you suggest.
Different languages may have different alphabets. For example, Welsh includes DD and LL as entities separate from D and L, as well as a few other digraphs, and does not include K, Q or Z. However the letter-forms are called "Roman", as illustrated here.