A painted bike line in Copenhagen is
huge compared to the lanes they paint in Britain.
The British version is about half as wide, and often allows parking for large parts of the day (i.e. is only a lane for a couple of hours).
I've never been to Oxford, but someone mentioned Abingdon Road [1]. Just some bike symbols painted on the road.
Or a little further up [2] -- that thing on the left is a British cycle lane. If you go forward 3 clicks, you can see it ends almost immediately.
You obviously wouldn't take small kids through crap like this: [3]. Unbelievably, this is "National Cycle Route 57" (the StreetView car is pointed the wrong way to see it, but if you search you can find the tiny blue "57" sign at the roundabout.)
Other than the bridges, the only part of Central Copenhagen with painted lanes I can think of is Bernstorffsgade [4].
I'm not really sure where you're thinking of for "lots" of bike paths. A little further out [5] is this, where at least the lanes go behind the parked cars, and are wider. https://osm.org/go/0NWvDIpQ-?layers=C shows solid lines for painted lanes, and there aren't many.
[1] https://www.google.com/maps/@51.7378951,-1.2521828,3a,70.1y,...
[2] https://www.google.com/maps/@51.7471822,-1.2563676,3a,75y,18...
[3] https://www.google.com/maps/@51.7504861,-1.2448391,3a,75y,12...
[4] https://www.google.com/maps/@55.6698747,12.5700607,3a,75y,31...
[5] https://www.google.com/maps/@55.6598222,12.6187002,3a,75y,16...