I would say it's definitely a phase for all kids - in fact, any person I've ever known *on the spectrum* tends to be "hyper-rule-oriented" at that age and prone to lie LESS than kids who are not on the spectrum.
Lying is actually very developmental. It's a "good" thing. Kids are learning fantasy from reality. They are learning what is important to you and that being "in trouble" for doing something "wrong / bad" doesn't feel good (and that they have the power to do something about it...... lie and they don't get in trouble. simple)
The best success I've had in dealing with lying starts with not asking open ended questions. If you KNOW he did it, then don't ASK him. That's a trap. Instead - "A, you have spilled your drink. please get a towel and clean it up". Then you merge into open-ended solutions. "what could you have done differently so that your drink did not spill?" It's more about him learning the PROCESS of HOW to tell you the truth, than about kids just confessing when cornered.
I also think kids at 5 or 6 understand that they are not telling you what actually happened, but they don't yet understand the "emotion" of a lie. *Lying* for most adults has an emotional / betrayal component that just isn't there for most kids. They simply WANT the situation to be different, or they don't want to get in trouble, so they say something other than what happened. It's not malicious the way lying can be as an adult. But we (as adults) feel the same impact.... which isn't on the same level as our kids. So, it's important to bring the lie down to their level...... which is very simple: they want their world to be different for that instance or circumstance or situation. They are not lying TO YOU. They are simply creating a different reality to avoid *whatever*.
I also think that 5/6 can be an age where you have to ramp back up 1:1 interaction. They could play solo at 4 and not get into trouble. But they're smarter now. Also going to school and getting some knowledge/experience that you, as their parent, don't always have knowledge of until it rears it's ugly head. YOU didn't teach them how to use the remote, but they may be using it at school, unbeknownst to you.... until they turn on the TV when you didn't even know they were up. That kind of thing.
One last thought.... my golden rule is to not get too worked up about it. All kids are going to lie. If you freak out, well then it makes it tough for them to come clean. If you keep your cool and keep the line of questions specific and pointed then it opens up dialogue.
You'll end up teaching him that it's better to come clean by making sure your reaction is always about fixing the situation. "I can't help you make sure all the milk is out of the carpet unless you come and tell me you spilled".