To specifically address your question as to whether the same collets are suited to lathe work and for holding milling cutters:
Ideally, lathe collets are longer in their grip length, whereas milling collets are kept short - partly because milling cutters have short shanks, partly to keep the overhang down.
However the only longer options are traditional collets, each of which has a very small diameter variation which it will grip (and damage happens if you try to grip smaller stock) OR rubberflex types (or Pratt / Burnerd or Crawford, "Multisize") which are rare and expensive.
In practice the ER series seem to work pretty well in a lathe. Possibly you'd go for tailstock support slightly sooner (with increasing overhang) than with old school.
The need for a small shop to get more bang out of limited bucks makes ER collets an amazing development. If you ever get into making up kit for tool and cutter grinding, they can serve there as well. They are a relative delight to use - one of those ideas which work as well in practice as they look good on paper. I was wary about cutter pullout for a while, being brought up on screwed shank cutters, but provided you don't buy rubbish, and torque them up properly, it seems they're reliable.
I had to send back the first holder I bought: the nut wouldn't pull far enough on to close the collets down to the bottom size, regardless whose collets I used. I'd specified I didn't want a Chinese holder, and was told this was Taiwanese, but I have several reasons for mistrusting that, and the vendor admitted it was hearsay only. They came up with a LAIP (Spanish) holder with a two-piece nut (aka bearing nut) where the nut has a built in plain thrust bearing, so the collet is not being twisted into a (VERY slow) spiral by the tightening torque. It's a really nicely made item, a delight to use, and I personally think it makes no sense buying collet gear on price alone. In their embarassment, they sold it to me virtually at cost, anyhow.
One major advantage of ER collet holders (at least the Morse Taper ones) over most other types is that you can bury a long shank inside the hollow arbor. What's more, the collets are happy gripping on the flutes of a drill. This means that your jobber length drills can double as stub drills of exactly the right length, often eliminating the need to spot drill or centre drill.
Can also be advantageous sometimes to put the drill in backwards: a quick way of setting up over an existing hole in the job, especially if it's small.
If you've got drill sets in 0.1mm increments (or the old letter and number series) then you've effectively got a graduated set of spigots/ locators for free.
If you do buy cheap ER collets, set each one up in your lathe headstock with a drill blank, piece of drill rod, dowel, spotting drill (of failing anything better, a standard drill backwards) and check the total indicator runout of each collet near top and bottom of its size range. (It was by doing this that I diagnosed the inability of the holder I had to pull down to bottom size) Should be 0.003mm max for fair to good quality, your call for cheap collets. Make sure your lathe taper is true before making a Richard of yourself, if they all seem to run out by the same excessive order of TIR.
Since seeing how spectacular the show is when a cutter pulls out I'm rather keen to avoid repeating that circus act!
