Is MacBook Pro 13 2015 able to run YouTube’s 4k 60 FPS videos?
I'm under the assumption that you may have stumbled across reports such as this one in a casual Google search. I quote:

The Air correctly identified the attached monitor at 4K resolution, switching the user interface into HiDPI mode.
Subsequent refresh rate tests confirmed that it was running at 60Hz, and the publication said that interface components were &quot;more than smooth enough for desktop use.&quot; Problems cropped up when performing animation-intensive actions, like entering full-screen mode, with &quot;clearly visible&quot; frame lag, showing that the display pushes the Intel HD 6000 graphics chip near its limit.

Notice that the first emboldened snippet says that you have to switch the user interface. This likely means usage of 3rd-pary software, such as SwitchResX, to enable changing system setting and entering this configuration. This is not recommended. Changing the resolution settings to something that cannot be enabled via System Settings can cause problems. Trying to push your hardware over the limits set by Apple can cause the system to overheat and consequently be damaged. Your MacBook is not ventilated very well and you have an integrated GPU which relies in part on the CPU. The additional overhead will cause complications.

I've read that full 4k at 60 fps is supported over DisplayPort. –  Šime Vidas 3 hours ago

Refresh rates in the article above confirmed 60Hz. Is it able to play videos at 60Hz? The answer is likely yes. It is officially supported? No, their support page shows a limit of 30Hz at the resolution you stated.
As per your quote, can DP support 4k @ 60Hz? Yes, the DisplayPort 1.3 standard supports it. Does Thunderbolt 2 support DisplayPort 1.3? No. Remember that through the Thunderbolt port there is backwards compatibility with a mini DisplayPort cable, but with Thunderbolt 2 there are still some limitations as it uses the DisplayPort 1.2(a) standard.

To summarize this:

Is the latest MacBook Pro 13 able to play such videos smoothly?

Objectively speaking, yes, it is able to.
Will the system play it at 60Hz on a 4k 60Hz monitor? No, it will not. It does not natively support it.
Also see DisplayPort 1.3 and Thunderbolt 2 on Wikipedia for sources and more detailed info.