Miniature wargames require miniatures, and the biggest wargame out there is Warhammer 40,000. The same company that publishes 40k also happens to manufacture miniatures (branded as Citadel). As a player and hobbyist painter, I appreciate the alignment between the miniatures and the rules. However, as a general wargamer, I also know that not all games have that alignment, and frankly 40k doesn't require it. The game mechanics work just as well whether you're using Citadel miniatures, Forge World miniatures, Wargames Atlantic miniatures, Lego minifigs, or paper cutouts. This prompts the question of whether there's any meaningful connection between Citadel miniatures and Warhammer rulebooks at all.
I think there are two ways to look at how Warhammer rules relate to miniatures. You can view miniatures as representations of a bundle of rules, or you can view rules as explanations of an extent miniature.
If you think of a miniature as a stand-in for rules, then any object on the table can fill that role as long as everyone at the table understands what it represents. As long as it's understood that a salt shaker, or paper cutout, or whatever, represents the attributes and values listed on the top half of page 153 of the Space Marines codex, then everyone at the table understands that its Movement is 6, its Toughness is 4, and so on, and that it has an Bolt Rifle and a Ballistic Skill of 3+, and so on.
The Thing that exists in this scenario are the rules. That Thing's proxy on the tabletop has no form of its own. In fact, it hardly matters what physical form it takes, because no matter what it looks like on the table, it's just a reflection of a collection of data.
If you see a miniature as a "living" entity, then that entity exists in your army. How effective it is on the battlefield depends on the data you get back from headquarters. Today, it's Movement 6, Toughness 4, with a Bolt Rifle and Ballistic Skill 3+, as listed on page 153 of the Space Marines Codex, and conveniently in Kill Team (2018) page 85. Tomorrow, it might be Movement 7, Toughness 4, with a Bolter and Ballistic Skill of 4, and so on, from Liber Astartes. Sometime in the future, it could be something else entirely, because you might use that miniature in yet a different game.
These two ways of thinking about miniatures and rules end with the exact same results: You put a toy on a table and then roll dice according to a set of rules. But the emotional relationship between that toy and those rules are distinct. They aren't exclusive of one another, though. We humans are complex, so you can maintain both views at the same time, even about the same miniature (although I think it probably varies more from miniature to miniature).
There are lots factors that influence how we think of a miniature.
There are more, of course. None of them are required to be rational.
The manufactured problem of a Citadel miniature either not existing (this happens in Horus Heresy, which used to survive almost exclusively on Forge World miniatures and whatever model a player decided to throw together) or being too expensive or being sold out or not being ugly, or whatever, seems pretty unique to Warhammer, as far as I can tell. Far more wargames have been and are sold, and played, with no associated line of custom miniatures. In most wargames, you see a miniature you think fits into the rules, and you buy it. Maybe you make a few modifications so that it's holding a pistol instead of a rifle, or a spear instead of a sword. Then you correlate it to some rules, put it on the table, and play.
Any tension around how a Warhammer army looks on the table, I believe, is down to an illusion that there's an external authority judging the validity or even coolness of your army, or that the intent of your army won't be understood by your opponent.
It's your army, so you can represent it with any miniature you please. This does carry the expectation that you're going to make a genuine effort to make the army communicate the rules it represents. If I'm playing a space marine army, then I'm not going to use my ancient Hattusia army, with their spears and slingshots. (I might, however, use my ancient Egyptian army to represent Necrons, in a pinch.)
I personally admire a Warhammer army that's not entirely "off the shelf". Most of my armies have at least one or two characters I've opted out of using a Citadel model to represent, or I've used a different Citadel model than the one in the books.
Horus Heresy has only recently started getting plastic models, and I avoid resin as much as possible. For most of the Cult Mechanicum army of the Heresy, I just use my 40k Adeptus Mechanicus miniatures. However, the Heresy books don't exactly have an abundance of images in them, so can be difficult to conceptualise how big a miniature is meant to be. Sometimes a resin Citadel miniature (formerly Forge World, I guess) is a great reference point.
The inverse can be true, too. Sometimes the Citadel model is a reference for what I don't want.
For example, the Nexos model for the Genestealer Cults in 40k happens to be one of my least favourite Citadel models. It's not bad, I just don't love that the miniature includes a control panel (drone? dining table?), so I decided to fashion my own Nexos out of spare parts.
The Vindicare Assassin miniature is another I don't love. The Assassin is fine, but I'm wholly averse to the stone pillar in the background. What if my Vindicare Assassin is in a metal bunker? Did he bring the stone pillar along with him? I couldn't bring myself to pay for it, on principle, and so I use a Wargame Atlantic miniature instead, holding a sniper rifle and painted black with a white face plate. It's easy, and in the end more satisfying for me to play with.
Other Citadel miniatures I would never swap out. Pious Vorne and really all the characters from Blackstone Fortress, my very first Adeptus Arbites, every single one of my Skitarii, my Kastelan Robots, Technoarcheologist, Enginseer (I guess most of my Adeptus Mechanicus, really), Magus, my Mark VI space marines, and, well, lots more, for lots of different reasons.
Sometimes the "wrong" miniature is exactly the right one for your unique army. Other times, the specificity of Citadel miniatures, having been fashioned exactly for the rules you're using anyway, are precisely what you're looking for. Whether Citadel models are a guide or a goal for you, I think it's fun to get creative with your wargame armies.
All images in this post copyright Games Workshop.