In the abstract, yes - unless something changes in your body (joint mobility, upper body strength and mass, power output etc), then for a given discipline of riding and with a given set of goals (all day recreational road ride vs racing, say), the fit should be able to translate between bikes.
Where things can go wrong or get murky has to do with the methodology of the fitter in a given situation. And, even if they do the best possible job, the new bike may not position you the same as the old one.
A fitter can use either a fit machine (aka sizing cycle and some other names) or an existing bike. A fit machine is a dimensionally adjustable stationary bike where all the contact points between you and the bike can be adjusted infinitely and independently. They could use either measurements from your body or an existing bike to get starting parameters, and then they'll adjust things until they like the result (hopefully you like it too). The best fit machines have resistance like a trainer and can also swap cranks and handlebars, so the pointed-forward position is virtually identical to what it feels like on a real bike.
With a fit machine, once you've achieved perfection there, you get a set of numbers that describe the spatial relationship of the contact points - handlebar geometry and angle, horizontal and vertical distance from saddle to bars (which is best done in reference to a specific saddle), saddle angle, saddle vertical and horizontal relationship to cranks, crank Q, crank length, etc.
Fitting in this way is good at conceptual purity and ideal for building a custom bike. It's also good at giving you numbers to go bike shopping with, since you can then use the manufacturer geometry charts to ensure you can get your ideal fit on a given bike, i.e. the top tube won't be too long or the highest achievable bar height won't be too low. (Using the numbers in this manner to their maximum extent can sometimes mean modeling the prospective new bike in CAD or on paper, especially if it's an edge case or if you're ordering a stem or other fit-related parts at the same time). In the same way, you might find that compared to your theoretically perfect numbers, the achievable fit is compromised on your current bike.
However, fitting with a fit machine is elaborate and time-consuming, and is not the only valid way of doing it, depending on the client needs. A lot of people want something more casual and inexpensive, which leads to the type of fit work that's done by adjusting parameters on an existing bike. If you do it this way, then no, there is no guarantee that what you come up with can be carried over to another bike. Some fitters who do it this way may have a method for being able to say or not say that the parameters attained are perfect and they wouldn't change anything even in the abstract on a fit machine, which is conceptually possible as a way of doing it but isn't what you'll typically find. Some will spell it out for you if they've found a limitation to what's possible on your current bike, but some may give you the best possible compromise and send you on your way. Doing it that way is practical for a lot of riders and situations and isn't a lazy method per se, but has major limitations when it comes to riders who are having fit issues they're having a hard time solving or have outlier/unusual circumstances.
If one way or another you do get into a fit on an existing bike you want to emulate on a new one, you can use those numbers in much the same way as a fit machine fit to go shopping for another bike based on whether the same fit is achievable.
