Can I call myself an assistant professor without a PhD
I think the problem here is that you don't get to choose what to "call yourself" (in professional contexts, particularly in your CV). Your job title is chosen by your employer. What you have control over is whether to take the job or not. If you took a job that has the title "Assistant Professor", then you are now an assistant professor, and it would be dishonest to represent yourself otherwise in a CV, email signature, business card, or similar professional communication. So yes, not only you can call yourself that, but in fact you cannot really not call yourself that.
At the same time, while your concern that this would look strange seems legitimate, to me it seems that the signal that you would be sending by presenting yourself as an assistant professor without a PhD is not so much that you are being dishonest or misleading, but rather that the institution you are working for has low standards for the professional qualifications of its professors. So perhaps I would suggest that when you tell people about your job, try to make sure that "assistant professor" is always accompanied by "[name of university]" so that no one has reason to suspect you of empty boasting. Hopefully if you explain your situation in a matter of fact (and accurate) way without appearing to be either boastful or apologetic, then instead of getting a bad impression, the person you're talking to will actually take away the fact that you are an industrious and ambitious person who is not only working his way towards a PhD in economics but also manages to have a steady full time job teaching at a local university while doing so. That's not too shabby if you ask my opinion.
Edit: to clarify, in casual conversation it is not entirely necessary to parrot the official job title. So for example as an answer to a question about your job, you can say “I teach at [name of university]” and not “I’m an assistant professor” if that makes you more comfortable. You probably shouldn’t use the word “lecturer” though, since at least in United States academia that means something very specific that doesn’t quite fit your situation.