In Megaman X7 (The slowest and weakest X game ever for some and WTF you can't play as X until you defeat all 8 Mavericks or rescue 64 reploids?!), everything starts with you playing as Axl and then Zero in tag team mode. So Megaman X7, Axl is introduced as an extremely cocky with a fondness for battle. So probably he was most likely based on Bass rather than Bass reconstructed. Or if he is Bass reconstructed, maybe the Red Alert found Bass' remains and created a new robot they named Axl. Maybe Red Alert then got Bass but wasn't able to find Treble, recreated him into the new generation reploid with no memory of the past but he does have some animosity with Megaman X in X7 and the two seem not to get along.
Now I think he's most likely a robot based on Bass. If we're to talk about gameplay, Bass in Megaman and Bass can fire in several directions but has to stand still and that was copied into Axl. Also like Bass, Axl relies on rapid fire bullets instead of charged shots. So pretty much if there's a difference to draw between them, Bass wants to beat Megaman in battle, Axl has no desire to kick Megaman X's ass (but best him out) but is always a braggart though. Also, Bass can double jump in Megaman and Bass but Axl can't, that trait was given to Zero. Also both names are based on sound.