I am looking for a wargame of any genre that has rules for creating stats for any miniature from any company. I know I've seen at least one set of rules like that, but I can't remember what the name of it was, nor can I find it through Google, as I'm not sure what to look for. It doesn't matter if it's a free game or if I have to pay for the rules, I just need a game where I can use the minis I've already got.
FASA's Vor was designed to do exactly this for generic science fiction - it's set in a huge void which sucks in planets and life from all over the universe. Unfortunately FASA's collapse prevented the detailed rulebook for custom miniature stats from ever being published, but the basic rules in the main set are workable, just not detailed.
Mechanically it resembled a conventional 40k-ish skirmish wargame, d10-based, with an interesting order system that allowed for good tactical use of cover.
(A new version has been in development, but it's Kickstarter project has just failed to raise the funds to get art and printing for the main rules printed, so I don't know if anything will happen.)
Aetherverse is also intended to allow flexible use of any force; this one set on multiple-parallel-Earths. I haven't played it, so can't comment on the mechanics.
Gruntz is a sci-fi, build-your-own-units/mecha/etc, miniature wargame. It's based on 15mm miniatures so that might not fit your desires, but you could possibly just double all of the length measurements if you're playing with 28mm/30mm minis.
I have not played Gruntz so I do not know how well it plays, is balanced, etc., but the one battle report I've read makes it look interesting.
There's a really old game system called Toying With Destruction by Guardians of Order (publishers of Big Eyes Small Mouth and the Tri-Stat system).
This system let you create stats for any type of toy at all, from a mech to a fire truck to a teddy bear. It had formulas to stat up all of these, and supported melee weapons, ranged weapons, movement, flight, ramming attacks, mounts/transports, and more. I believe it required a ruler and some D10's. It was simple but fun and very versatile.
Sadly this was only published as a stapled pack of paper, and was out of print a long time ago, and Guardians of Order closed its doors in 2006. Here's an archive of the game info, and another page with information and a cover picture.
We've used this for a number of our games Nice rules set and very customizable
Quite a few, actually, have had such a "Look at it and rate it" approach.
Early Warhammer games
The first edition of Warhammer 40K (Warhammer 40K: Rogue Trader), and the first several editions of Warhammer Fantasy Battles, were all intended to use any minis, and had extensive minis rating lists by base type, weapon, and profile value.
40KRT vehicles were points-costed by their ratings; the rating system wasn't terribly effective at play balance, but it was workable within reason, and especially for man-sized units.
Ground Zero Games' games
GZG's rules are intended for use with both their lines and others' lines of minis. Better still, you can get them in PDF for free direct from GZG.
Dirtside II is vehicle or squad per mini, platoon scaled, up to battalion level. It includes vehicle design rules that border on being "Look and rate"...
Stargrunt II is tactical, squad level, individual man per mini, infantry combat, but it also can use Dirtside II vehicles.
Full Thrust is space combat, one ship per mini. It can do fleet level play easily.
R Talsorian Games' Mekton
Mekton has always been a combined battle game and RPG, and always had a design system capable of producing a wide range of designs.
Because the design system is actually a design system, you might not always be able to get decent match to a given mini's fluff text.
7th Street Games' Mecha
The system is able to use a variety of minis. The included setting is less flexible, being focused on space aztechs.
Dream Pod 9's whole catalogue
The "rating" system in both Heavy Gear and Jovian Chronicles is intended to take an extant mini or image, and allow you to devise game stats to fit, generating a relatively good point total for balance purposes. Other games in the line use the same ratings, so that means that any of them can be used. Several have both minis and RPG rules, and the minis rules are almost all written for hex-grids but easily adapted to gridless play.
Flashpoint! is a wonderful minis wargame, using a mostly 1d20 rolls, printed in landscape mode in the 90's, of skirmish scale combat in a near-future sci-fi setting. 1 Character/vehicle/robot per model, typically individual model maneuver scale, and intended for platoon level and below.
Many elements border between a rating system and a design system, especially robots and vehicles.
Old FASA games
VOR is very similar to early WH40K:RT... except using d10's rather than d6's, and having a less-obviously Traveller/Cthulhu mashup setting. The rating system is very similar to that in 40K:RT.
And onto the less close fit to the question...
Battletech, while having a design system rather than a rating system, has rules for mecha, vehicles, and troops, and multiple games in various scales, so you can use pretty much almost your entire collection somewhere... and it's still in print, from Catalyst.
Renegade Legion is very much the same approach as Battletech, but is a different set of combat mechanics.
I'm not massively familiar with wargames, but I thought this suggestion from another question may be of interest to you.
Brikwars is a gaming system intended to be played with Lego-esque figures. However, the stats for each figure are largely based on the figures dimensions (e.g. rules about weapons), and so it is possible to create stats for any figure of any kind.