Depends on the area of asset management you're going for. Alternatives will be tougher, especially straight out of uni. Wealth management is less competitive, though you'll have to be more polished I'd imagine. The more socially awkward will find a better fit in IB.
The best degree is soemthing you're good at and enjoy. You can always add extracuricular and some exams / certifications to show finance interest / qualification (e.g. IMC, there's also ones for wealth management from the CISI I beleive). Having a non-finance degree may actually be more beneficial if you're in a client facing role in WM, as you'll have more to chat about with clients who likely didn't do finance.
No, for asset management you can also pretty much say fund manager, i.e. they manage funds for private and/or insitutional clients in funds, which are seperate legal entities. The clients are mixed, but for msot large fund managers, insitutional clients will be making up the bulk of the money. Examples are Blackrock, MAN Group, HSBC Global Asset Management, UBS Global Asset Management.
Fr wealth management ,think more along the lines of private banking. Clients are usually wealth private individuals who keep the majority of their wealth with you. Examples are Coutts, Julius Baer, HSBC Private Banking, UBS
Some banks are also mushing them together a bit more and start to blue the lines, e.g. Deutsche Bank Asset & Wealth Management or Barclays Wealth and Investment Management. A lot of smaller firms also mix it up, e.g. Ruffer, Close Brother etc.