Of course Japanese coworkers can become real friends outside work. There is no strict rule that says coworkers belong in one category and friends in another. You see each other again and again, you discover that you get along, you exchange contact information, and at some point you simply become friends.
What happens after that depends on the people involved. Some go out for drinks after work. Some occasionally have lunch together. Others usually meet as part of a group. There is no rule saying you must spend your days off together to count as real friends, or that a friendship was not genuine if you lose touch after one of you leaves the company. That has more to do with personality and the relationship itself than with anything uniquely Japanese.
One thing that can be confusing is that even close friends may seem more distant at work. There are bosses around, differences in rank, and sometimes clients or business partners watching. Two people who speak casually outside work may use polite language in the office and treat each other much like any other coworkers. That does not necessarily mean they are less close. Sometimes they are simply behaving as people are expected to behave at work.
With foreign coworkers, there can be one extra barrier at the beginning. Someone who has little experience interacting with foreigners may start worrying: “What should I talk about?” “How do I invite them to dinner without making it awkward?” “Will the unspoken understanding that works between Japanese people work here too?” They may never have planned to study abroad, but suddenly it can feel as if the study-abroad experience has come to them at work.
So if someone seems a little stiff at first, or turns down an invitation once, that does not automatically mean rejection. After a few conversations, once both people become more comfortable and realize they get along, the distance can shrink very quickly. Someone who seemed guarded at first may turn out to be very kind and surprisingly easy to get along with once you get past that initial barrier.
There is no special rule for becoming friends with a Japanese coworker. For people who are simply not used to interacting with foreigners, the first step may just take a little longer.
