Site search certainly produces "result pages", so it all depends on whether you not you think your site search is powered by a "search engine".
Most of the definitions of "search engine" on the web are broad enough to include site search. They define a search engine as software than searches documents.
a computer program that searches documents, especially on the World Wide Web, for a specified word or words and provides a list of documents in which they are found.
computer software used to search data (such as text or a database) for specified information
Search engines are programs that search documents for specified keywords and returns a list of the documents where the keywords were found.
Under these broad definitions of "search engine" your site search results could easily be called "search engine result pages" or SERPs.
There is also a second definition of "search engine":
also: a site on the World Wide Web that uses such software to locate key words in other sites
A search engine is a web site that collects and organizes content from all over the internet.
If you think of a search engine as a website that provides search results for other sites, then it would seem odd to include site search results in the term SERP.
Your colleague is not wrong to use SERP for site search results based on the broad definition of "search engine" meaning software program. However, I would not use the term myself for it because so many people would think like you and be confused.
In fact, I rarely use the term SERP at all. I prefer just calling them "the results" or "the search results" which is short enough to say and doesn't require unfamiliar people to learn new jargon.