Every company these days is a tech company in some respects, but I don't think Tesla fits the bill in the general sense. Tesla is a green energy manufacturing company.
The only reason I think this matters is their finances are a lot different than a tech company. You are correct that Tesla is currently priced like a tech company though, so maybe I'm way off!
"Technology ("science of craft", from Greek τέχνη, techne, "art, skill, cunning of hand"; and -λογία, -logia) is the collection of techniques, skills, methods and processes used in the production of goods or services or in the accomplishment of objectives, such as scientific investigation. Technology can be the knowledge of techniques, processes, and the like, or it can be embedded in machines which can be operated without detailed knowledge of their workings."
You can argue that that isn't accurate, but I don't see how you can argue that the categorical statement "tech companies have server costs" follows from it (I can't even find the word "server" on that page, or on https://en.wikipedia.org/wiki/Tech)
Yes, but even they can't keep up with the projected demand for batteries which is why Tesla is building out its own factory [1] in partnership with Panasonic [2] to manufacture batteries.