My first modern bike. Can somebody tell me what kind of bottom bracket is this? Octalink? Cuz the seller told me it's cup and cone.

Ritchey compact crankset 50/34

Yes, cup and cone, not Octalink or ISIS. So probably either a ball-bearing retainer design, or loose bb's.

Old school cup and cone. Serviceable and reliable enough. Can be replaced with a cartridge unit.

The kind I like.

THe spindle likely uses the older squate taper BB/crank interface. If you remove the crank-fixing bolt you can see the end of the spindle and figure out for sure what type it is. Be sure to firmy re-tighten the bolt before riding.

Why do you ask, does it need replacement?

It feels like my chainline is kind of messed up. Ritchey compact double (50/34) and 10 speed 105 cogs (12-27).
Pedaling in small chainring and largest cog, large chainring would occasionally rub and catch the chain. Not sure if it's an adjustment issue or not. Quick measurement shows that it might be 2mm off the presumed 43.5mm chainline.

They don't make bottom brackets in 1mm increments. If you're only 2mm off from the ideal, then you're about as close as you're going to get. Most likely, you'll just have to tweak either the limit screw or the cable tension.

It is hard to tell if this is the case for the OP, but there are some BBs that can be adjusted by loosening lockrings on both sides and threading the BB to one side or the other. However, if the pedals are close to centred in the frame, 2mm will not make enough af a difference (and is likely within most peoples' ability to measure chainline) to go tot he trouble.

Also, that is kind of a weird setup - unless it is an aftermarket boutique BB, you normally only find cup-and-cone BB shells designed for splined BB tools on the cheapest of the cheap bikes. I suspect there is either a parts mismatch (cups from one style of BB kludged to work with another style), it is not a cup-and-cone BB, or someone has gone out of their way to take an ultra-cheap cup-and-cone BB from a X-Mart bike to install on this (apparently) decent aluminum road bike. Or perhaps it is a BB designed for a 73mm shell and the lock ring was added to cover up the extra exposed threads.

The chain should be nowhere close to the big chainring when you are in the small ring and the largest cog. Do you mean that you are in the smallest cog in the rear, and so the highest gear available when in the small ring? That's a common problem due to the size difference between the 50T and 34T chainrings, and one of several reasons you should be in the larger chainring at that point.

Not likely on a cup & cone type bottom bracket. But looking at the OP's picture, if the problem is that the chainrings are too close to the frame, there does appear to be room for at least a 1mm spacer under the fixed cup to move the arm out. If the problem is that the chainrings are too far away, the OP would need a shorter spindle.

I meant when I'm in 34 front and 12 back. I know to avoid this combo, but just think it's kind of odd that this is a common problem with compact doubles.

I have serious doubts that this is a cup&cone BB.

I don't think this is unusual and this combination is discouraged on compact setups. Whether or not you get an actual rub is a function of the chainline and chainstay length.

The idea with compact doubles is that the big ring is used with the whole range of the cassette for most of your riding, and the small ring is used with the bigger cogs when you have a hill to climb. It's been common wisdom throughout the decades to avoid the big-big and small-small combos, but with today's flexible chains it's more of an annoyance.