There have been many studies conducted examining the similarities and differences between L1 (first language) and L2 (second language) acquisition. Unfortunately, it is difficult to accurately measure a learner's lexical size and their depth of knowledge of individual words and many researchers disagree as to how best measure those 2 aspects.

Jiang in a 2000 study presents 3 stages of development.

1) Strong link between the new L2 vocabulary and L1 translations.

2) Conceptual mediation stage where conceptual meanings of words are mediated with L1 translation equivalents to make up for a lack of links between L2 meanings and concepts.

3) Conceptual links are established for L2 words giving them the same status in the learner's mind as L1 vocabulary.

So, while some similarities may occur in the developmental stages between learning words in L1 and L2, one key difference is when learning L1 there is nothing to compare words with all fall back on. When learning an L2, learners may initially rely on their L1 knowledge, especially when trying to link words with concepts.

Source is Rod Ellis, "The Study of 2nd Language Acquisition". Am currently completing my masters in applied linguistics.

Also, I speak Japanese and hold a firm belief that the level similarities and differences between a learner's L1 and L2 (e.g. a French person learning Italian - both romance languages versus an English speaker learning Japanese - totally different languages) can greatly affect the process of L2 acquisition.

Although some researchers argue most L2 learners undergo similar acquisition stages and processes, there also exists plenty of research suggesting otherwise.

Also, acquisition of lexical items (words) is very different to, say, acquisition of grammar, phonology etc.