Yes, it relates to major league time.
3 years at league minimum (contracts auto-renew)
3 years arbitration-eligible
Of course, you can "buy out" some/any of those years by negotiating a contract extension. I generally only do that when players are arbitration eligible because (IIRC) allowing players to go to arbitration can affect their mood and willingness to negotiate with you down the road. At the very least, it's very common practice in the real world to sign players to 1-year contracts rather than let them go to arbitration so I tend to follow suit. The AI generally does not do this.
After that, the player is eligible for free agency. There are some exceptions to that (namely, "Super Two" players who become arbitration eligible a year early).
Wikipedia breaks it down pretty well:
https://en.wikipedia.org/wiki/Major_...ry_arbitration
But just because they never see the majors doesn't mean they can't become free agents - after a number of years (I think it's 6, maybe 7) minor leaguers can also become free agents, although you can usually re-sign them to minor league contracts and they stick around.