Can I call myself an assistant professor without a PhD?

the position is called "Assistant Professor"

If you hold an appointment as an assistant professor, then you are an assistant professor. Still, I understand your concern....

kinda makes no sense due to the size of the institution and the lack of research required by it...however this is a bit of a no-name brand institution

Context is everything.

• I agree it would look foolish or pompous if you use your title in a way that gives the impression you are a professor at an R1. When meeting colleagues at different institutions, you may wish to describe yourself as a lecturer or say "my title is assistant professor, but it's mostly teaching" or something unambiguous like that.
• On the other hand, everyone at your home institution should understand what an "assistant professorship" at that institution entails, so there is no issue with you using the title internally.
• Similarly, non-academics don't know or care about the subtle differences in academic rank, "I'm an assistant professor" is fine in such contexts.

Regardless, your PhD (or lack thereof) is irrelevant.

Sadly, such overloaded terms in commonplace in Anglophone academia. Many of us with doctorates select the salutation "Mr./Ms." to avoid being confused for an M.D. ("real doctor"). Those with PhDs from online schools face a similar dilemma.

Academic titles are not something you award yourself. In the US, Assistant Professor has a specific meaning and is assigned by a university. Other places it may have no meaning at all.

But, I think people would react poorly to you doing that. Your title is Lecturer, as you state. Use that, and only that until you are awarded a different one.

Even if you have a PhD you aren't an Assistant Professor unless a university says you are. And if you leave that university you lose that title. Caution is advised.

Since the question was edited, just ask the institution what you can call yourself. But note that the title doesn't carry with you and other institutions might not accept it. Again, caution, especially in job applications and such.

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.