That's an EA888 engine you have, and it's a 2012 so it's the VL or "Gen 2" variant. Your engine code should be CESA or CETA. You have what is nearly identical to a CCTA or CBFA motor (the Gen 1/non-VL) but you don't have a MAF sensor. (Actually it's possible even the CCTA/CBFA didn't use a MAF on the TT but I'm not 100% sure on that.)
The problem with that Uni intake you pictured (UH002-INA) is that it says for "TFSI" models which is rather ambiguous because Audi used TFSI and "Turbo FSI" terminology all over the place, regardless of the engine type. However for most VW-related badging they call the older EA113 engines "FSI" while the newer EA888 engines (like you have) were called "TSI". You can see your engine cover says "TFSI" as many Audi 2.0Ts do, regardless of what engine family they are, which I can see makes it confusing.
So the intake you pictured is actually for "FSI" or EA113 engines, which is not what you have and hence why it doesn't fit. It also has a MAF drop-in on the top pipe there, which would be another indication that's not the right one.
So really what you want is a "TSI" intake, preferrably one for your particular engine w/o the MAF. The other ones listed for CCTA should fit but you'd have to account for the absence of the MAF... The Uni one which would fit better (but still not 100% correctly out of the box) would be the UH001-INA model. You can see the elbow that goes on the filter and the main pipe are separated. The idea is you're supposed to insert the factory MAF housing (which you don't have) in between those two parts. You'd have to connect those together, possibly using a pipe or tube of the same diameter and tightening down the clamps. If you go on Uni's site you can see they don't actually list a compatible intake for your car, which basically comes down to the missing MAF.
So going from there if you go with intakes that don't replace the main pipe that goes into the turbo, you shouldn't have too much issue fitting them because now the MAF doesn't matter. So the APR and Neuspeed ones should fit, going by what ECS has on their page. The APR ones (PEX and CF are the same just different material and price) you'd need to buy the bracket for the TT, as mentioned on the page.
TBH I would just forget about the intake altogether. It's not gonna do much of anything for performance and is just going to make more noise. Of course if noise is what you're after, that's another story. But for the cost they're not really worth it IMO. For performance best thing to do is just take out the snow guard from the airbox and have a clean filter. I'd save your money for a tune and downpipe, if you don't already have those things--those have a much greater effect on engine power.
