I know that cross-generation (PS3 player with a PS4 player) is not supported but can a XBOX One player play together with a PS4 player?
As far as I know, no it is not cross platform. Bungie stated it is due to 'fairness' as the consoles (/ pc if they added it) would all be slightly different in performances.
Destiny won’t feature any sort of cross-platform play [...] This isn’t due to a technical issue though, but rather Bungie not wanting to give players on the new consoles an unfair advantage, as Bungie Engineer Roger Wolfson told Digital Trends:
I’ll speak for the hypothetical player. I have a disadvantage sniping across the map because [my opponent with a new-gen console] is only two pixels on my screen and I’m four pixels on his. You see that in the world of PC gaming, where people are always racing to the best video card to give themselves the advantage.
Regardless of where the reality is, there’s definitely a perception among gamers that better hardware means you have an advantage. We don’t want to have to enter that fray, so to create the best, most level playing field, both actually and perceptually, we separated it by platform.
The above is the reason given in the linked interview, by a Bungie engineer - though there are as stated in the below comments other reasons for why it is hard to cross platform a game in general. For example SONY and Microsofts platform policies / ability to work together.
No, this is not possible, unfortunately. See this post for an earlier similar question. Here I explain that the servers are seperater for each console, but you can transfer your profile cross generation, but not cross system (Ex. from PS3 to PS4, but not PS3 to Xbox 360/One).
This answer is entirely speculation, but as a software developer myself I feel it is educated speculation.
The biggest reason that I can come up with for Bungie (or any developer really) not to do this is that there is no framework in place for cross platform matchmaking, within each console the devs don't need to do much work (if any) to tap into a network of players for matchmaking. In order to implement a dedicated system to do so for the game would balloon both the cost of making the game and the time it takes to deliver the game (and keep in mind Destiny was already under development for more than 5 years and cost over $500 million).
Along side the lack of a shared network of players, they would also need to implement a custom friends system.
And the final reason that I can attribute to this goes along with their published reason for no cross generational play. These two systems are not exactly the same when it comes to hardware and although technically it won't make much of difference (if any) there are very many people that would use this difference as an excuse and flame the life out of Bungie.