I'm not a historian, but a marketing academic, and I will try to speak to this question from that perspective. At a base level, this is a question of the effect of product placement. There is ample evidence for the effects of product placement, in the forms of increased recall and behavioral change (i.e. buying the product).

Addressing the Frasier-Starbucks placement directly, I poked around a bit and there are some academic papers that have tenuously made this connection, usually in a speculative or hand-waving fashion (Russell and Stern 2006, Williams, et al. 2011). However, the Williams, et al. cite seems to be an example of an "academic urban legend" (per one of my favorite papers, Rekdal (2014), where the authors of the later paper cited something without carefully examining the original sourcing of the claim in the Russell and Stern paper, which appears to be Tueth (2000). The Teuth paper doesn't make this claim at all, and is instead about how 1990s situation comedies were frequently set in urban environments, such as New York City and Seattle and the cities become central elements of the plots.

That said, the lack of any empirical research testing your question doesn't really provide an answer to it. Addressing it would be quite difficult, though. The question as posed has an underlying causal claim - Frasier became popular, and the importance of Starbucks within Frasier led the viewers of Frasier to want to go to Starbucks and this increased its popularity. Supporting this, we have data that shows that concurrent with Frasier going on air, Starbucks had a large scale expansion - a correlation. But, as they say, this is not sufficient for us to make a causal claim. At least two other possibilities would also fit this data: that Starbucks made Frasier popular ("backwards," as you note), or that the forces driving the popularity of Frasier and the opening of Starbucks stores are unrelated but occurring at the same time. Teasing these processes apart is not trivial, and making a causal inference based on observational data at a macro level like this is a common problem in economics, political science and sometimes in my field as well.

Fundamentally, this involves identifying some kind of counterfactual which we can compare with what we observe. One potential way this could be down would be to identify if there are communities that were not shown Frasier that we could compare against those that did have access to it, and see if these areas are less likely to be Starbucks outlets in those areas that were not exposed. It looks like the first reference to Starbucks in the show was in season 1, episode 2, where Frasier says "Hey, this isn't my coffee. Where's my finely-ground Kenya blend from Starbucks?" which first aired on September 23, 1993. Unfortunately, most of the media markets in the United States appear to have had NBC affiliates as of 1993 (per Wikipedia's list), and those that didn't likely could get access from neighboring broadcast affiliates. Moreover, those markets that didn't have access are likely to be quite different from those that did - for one, they're smaller markets, and thus would be unlikely to be places where Starbucks would open anyways, so this won't help us differentiate a causal effect of Frasier from unrelated drivers (i.e. having a market of sufficient size).

There are certainly other methods that might be used to demonstrate this, but its unlikely that we'd be able to show anything that will definitively point at any of the three different causal pathways. As with any large scale phenomenon operating at the intersection of social forces and popular media, its probably a bit of all three. Starbucks and other espresso bars were already becoming reasonably popular in the United States prior to Frasier airing. See this 1992 NYT by Fabricant article, introducing the masses of NYC to "something called espresso."

This new coffee culture is catching on in New York, where pots of 90-cups-to-the-pound weaklings sit on warmers for hours on end. Virtually every New York restaurant, from the exalted to the expedient, now serves something called espresso, strong Italian-style coffee in small cups. In Manhattan alone, the number of new espresso bars, cafes and stands that offer hot and cold, regular and decaffeinated espresso-based drinks with names like caffe latte have risen in the last year, from only a few to more than a dozen.

Most likely, the writers of Frasier identified this cultural phenomenon that was already taking place, and picked a brand that would be known to some of their audience but not to others, providing an "in joke" or reference that would require some cultural capitol to understand, especially prior to the widespread growth of the internet. Bringing this to a wider audience likely positively impacted the popularity of Starbucks - 27 million people viewed S1E2 of Frasier, so it would be surprising if it didn't. However, I think that breaking down attribution of the growth of Starbucks to references on the show vs. the broader societal interest in coffee is probably not possible.