This may be considered off topic given that it's kind of a gear recommendation. But I'll give the general concept of multi vs stompbox a go.
Firstly, any time one thing is emulating another, it will never be a perfect emulation. It may be better or worse according to personal preference, but it will probably be different. And even among dedicated pedals that are supposed to do the same thing—even two that emulate the same circuit—there will be vast differences in opinions from person to person.
So if you want a specific sound, probably try to get the thing that makes that sound. But if you aren't after a particular sound—say you just know you want certain effect types like overdrive, wah, delay, etc—then just try the cheaper/convenient option first and see if it works for you. You won't know if you'll like it until you try it.
That said, along with the emulation point, there's another broad generalization about the economics at play. Any time you get a bunch of stuff for a lower price, that probably means that there are compromises being made somewhere to hit that price point. In the case of a single dedicated pedal, the designers are probably spending more time on that particular effect whereas with a multi-effects unit the designers may be spending less time on each individual effect or feature in order to keep the cost down. That doesn't mean you shouldn't buy it though. If it sounds good to you then why pay more?
Are these multi-effects pedals digital (in other words like the
difference between a valve and a solid state amp)?
Basically yes they are digital. Sometimes there are pedals that might include several analog effects in one pedal, but multi-effects units like a Zoom are usually completely digital. Some dedicated pedals are digital too and it's not necessarily a deal-breaker.
But the tube/solid-state analogy isn't really apt. Both solid-state and digital can be fine and aren't inherently bad. But solid-state amps rarely emulate tube amps well. Digital pedals (or software more generally) can typically emulate analog effects much better than solid-state can emulate tubes.
EDIT: followup to your comment
I mean obviously there is a reason people pay $1500.00 for effects as opposed to $200.00.
The number one reason in my book, which I sadly left out until now, is the modularity. You can pick your favorite sounding pedal for each effect. Or you can still skimp and buy a cheap pedal if you're not that concerned about that particular effect. But you have the choice.
I do wish the answer delved more into the types of pedals (Digital/SolidState/Analog/Tube?) and what each type is targeted towards.
I wouldn't worry so much about the circuitry rather than the sound. But that said most pedals are either solid-state analog, digital or some combination of the two. Pedals with tubes exist but are relatively rare. Things like overdrives tend more to be analog while some time-based effects, especially reverbs, tend to be digital. Generally the more a pedal does or the more dramatic the effect the more likely it might involve some digital processing simply because it may be hard to recreate that effect with analog circuitry alone and especially at pedal scale.
For example reverbs are typically digital because in the analog realm you're pretty much limited to either spring, plate, or something closer to a delay and springs and plates take up too much space for a pedal.
Some, like a delay, could go either way where the analog ones tend to accurately recreate the sound of the era before digital delay (ex 70's bucket-brigades) but they tend to lack features compared to digital delays.
Sweetwater has pretty good filtering for this kind of thing. You can drill down to each type of effect and then use the filters on the left to see only the analog or digital ones.
All other things equal I tend to prefer keeping my signal path analog, but I wouldn't shy away from a digital pedal based on that alone (I mostly prefer my amp's spring reverb but do have a digital reverb and use a Boss multi-effect sometimes).