it’s along process costing lots of money.
if not already a company with published tiltes on the platform when you approach sony or ms or at least published titles.
you’ll need such things as
proof of company viabitlity-have you got the cash
proof of technology-usually dog and pony to show you can make a game
submission of concept-idea for game with info on engine specs, art specs, game design…
which must be signed off on by sony or ms who may require changes before approval
submission of game team-doc on your team(skills, education, past projects…)
plus many more steps.
dev kits first go to internal teams, then 1st party major developers with previous license…
on down the line with small start ups way at the back of the line (unless you really can impress them with the above mentions items so they just have to have your game). normally expect a wait of a year or more after a new system is introduced before they get to you.
Microsoft at one time had a program to help small developers to put games on the xbox, mainly due to the fact when it first came out a number of companys weren’t interested in making titles for the system. they might still have such a program, you might check.