Mustang will fit most traditional music styles well (might not be the ultimate death metal bass or slap bass, but basically wherever a P bass would fit). Besides the name recognition, the tone is the reason Fender basses are so popular - the tone just seems to fit right.
Speaking of Precision basses, that would be a good one to consider too. But the Mustang with its short scale is easier to pick up and play, especially coming from a guitar perspective.
Meanwhile, you seem intrigued by the Embassy Pro, so that could be a good way to go, too. Basically, the one that gets you jazzed (oops, did I say Jazz?) to pick it up, practice and play and learn is the one to get. If the Embassy Pro is tripping your trigger, go for it. It’s affordable, the people who buy them seem to like them (can’t recall anyone saying they didn’t like theirs), it’s a somewhat unique classic , and if it turns out it doesn’t fit everything you end up wanting to do, it will probably still have its niche down the line - could be a lifelong keeper.
If it were me, I’d get a Mustang (heck, I did - my MIJ Mustang is a lifelong keeper), but I’ll admit that checking this Embassy Pro video out just now has me kinda G.A.S.ing for one a bit, lol:
Just not sure I need another long scale bass (except for my Precision, I’m leaning short scale anymore), but those pickups are intriguing and the Embassy Pro body would seemingly eliminate the neck position/diving issues of the Thunderbird. Could be a cool Gibsony complement to my little herd, who knows. Basically an old school Thunderbird, except in a body that makes ergonomic sense.
Well, it looks like we have the same problem - too many intriguing choices!
Good luck. Either one should be good to get started on! Eventually, after learning to play and gradually trying out basses at stores and friends’ basses, etc., you’ll start to find out what works best for you/what you want in a bass. For now though, the main thing is simply to get started on a decent instrument (everything works, can play in tune and hold a comfortable setup) that gets you inspired. Whichever one trips your trigger is the one to get