AMD Equal to Intel i3 6100?

It depends quite a bit on the game you're looking at:
-- For Honor (http://www.techspot.com/review/1333-for-honor-benchmarks/page3.html): the closest is the FX-9590. For some reason (possibly the older game engine), For Honor doesn't give a lot of love to FX chips.
-- Gears of War 4 (http://www.techspot.com/review/1263-gears-of-war-4-benchmarks/page4.html): FX-8350 is pretty close in performance.
-- DOOM (http://www.techspot.com/review/1173-doom-benchmarks/page5.html): Again, you'll need an FX-8370 (or probably 8350) to be right behind the i3-6100.
-- Battlefield 1 (http://www.techspot.com/review/1267-battlefield-1-benchmarks/page4.html): You'd need at least the FX-8370 (or probably 8350) if using DirectX 11, but with DirectX 12 all you need is an FX-6350.
-- Deus Ex Mankind Divided (http://www.techspot.com/review/1235-deus-ex-mankind-divided-benchmarks/page5.html): FX-8350 is right behind the i3-6100, but the FX-6350 is right behind that one.
-- Titanfall 2 (http://www.techspot.com/review/1271-titanfall-2-pc-benchmarks/page3.html): As long you have at least an FX-4320 (i.e. 2C/4T CPU), you'll match the i3-6100 in performance...although the Athlon X4 860K isn't too far behind.
-- Tom Clancy's The Division (http://www.techspot.com/review/1148-tom-clancys-the-division-benchmarks/page5.html): You want at least an FX-6350, although the FX-4320 isn't too far behind.
-- Rise of the Tomb Raider (http://www.techspot.com/review/1128-rise-of-the-tomb-raider-benchmarks/page5.html): As long as you have an FX-4320, or even the FX-8320e 95W chip, you'll match the i3-6100 in performance.
-- Overwatch (http://www.techspot.com/review/1180-overwatch-benchmarks/page5.html): Again, as long as you have at least a 2C/4T FX-4320, you'll match or exceed the i3-6100; the Athlon X4 860K didn't play as well, however.
-- Dark Souls III (http://www.techspot.com/review/1162-dark-souls-3-benchmarks/page5.html): Technically, you'd want at least an FX-8350, although the FX-6350 was right behind it. However, that was tested at 1440p, because (as indicated in the article) the game is capped at 60FPS. Since even the FX-4320 made it over 50FPS with a GTX 980Ti at 1440p, I strongly suspect that at lower resolutions you'd be hard-pressed to see any difference between it & the i3-6100.

That's just games released within the past 12 months. Going back too much farther than that, & you have to start interpreting from the results because they weren't tested with the i3-6100. If I had to guess, though, it would probably be something like this:
-- Fallout 4 (http://www.techspot.com/review/1089-fallout-4-benchmarks/page5.html): Since they actually tested with an i3-6100, you'd need an FX-9590 (or a heavily OC'd 8350/8370) to approach the i3's performance.
-- Project CARS (http://www.techspot.com/review/1000-project-cars-benchmarks/page5.html): Although the game engine loves Intel CPUs more than it does AMD CPUs, it also loves physical cores much more than HT cores. Based on how the i3-4130 performed, I don't see the i3-6100 doing much better (if at all) compared to an FX-8350.
-- GTA V (http://www.techspot.com/review/991-gta-5-pc-benchmarks/page6.html): Again, based on how the i3-4130 performed, & how increased clocks didn't really affect the i7's performance, the game really prefers physical cores to HT cores. So a hexa-core FX chip should be more than sufficient to match the i3-6100, if not an octa-core.
-- The Witcher III (http://www.techspot.com/review/1006-the-witcher-3-benchmarks/page5.html): You'll need an octa-core FX chip to match the i3's performance.
-- Star Wars Battlefront (http://www.techspot.com/review/1096-star-wars-battlefront-benchmarks/page3.html): Again, very little effect from higher clock speeds on Intel chips, so an FX-4320 (or even the Athlon X4 860K) is enough to match an i3-6100 in the game.


That being said...

PCPartPicker part list / Price breakdown by merchant

CPU: Intel Core i3-6100 3.7GHz Dual-Core Processor ($108.37 @ OutletPC)
Motherboard: Gigabyte GA-H270-HD3 ATX LGA1151 Motherboard ($101.98 @ Newegg)
Memory: G.Skill Ripjaws V Series 16GB (2 x 8GB) DDR4-2400 Memory ($96.99 @ Newegg)
Total: $307.34
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-02-22 14:04 EST-0500

PCPartPicker part list / Price breakdown by merchant

CPU: AMD FX-8320 3.5GHz 8-Core Processor ($130.98 @ Newegg)
Motherboard: Gigabyte GA-990FXA-UD3 R5 ATX AM3+ Motherboard ($110.05 @ Amazon)
Memory: G.Skill Ripjaws X Series 16GB (2 x 8GB) DDR3-1866 Memory ($109.77 @ OutletPC)
Total: $350.80
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-02-22 14:05 EST-0500

The AMD build technically looks about $50 USD more expensive. BUT, you get the following:
-- an unlocked CPU (i3-6100 being a "locked" chip), &
-- a motherboard that you can actually overclock with (H270 boards can't overclock, you'd need a Z170 or Z270 board, Z270 if you want to be able to later switch to a Kaby Lake or later CPU). Going with a Gigabyte Z270 board that has a built-in M.2 slot for an SSD, for example (https://pcpartpicker.com/list/nDqfRG) boosts you up to $370 USD, or about $20 more than the AMD build.

So...from a performance & a cost perspective, you can build either an i3-6100 system or an FX octa-core system.

But I wouldn't recommend the FX system.

Although I'm an AMD man myself (as you can see from my signature, where I have the same motherboard as listed in the sample build), it's a "dead-end" system right now. It's not that the system is bad, per se...you're just limited on the future upgrades you can do (kind of like with any Intel Haswell build). Plus, Ryzen is right around the corner (pre-orders apparently starting today, with release set for 3/2/2017), so if you're looking to build a new system I would strongly recommend waiting another 3-4 weeks. But if you already have the FX-based system, don't worry, you already have a system that will perform as well as an i3-6100 system.