How to make less show the filename

Use less --follow-name if your version of less supports it.
That option was introduced in version 416.

Then do a normal follow command Shift+F within less.

Not a task for less

No,
I do not think you can do that directly, because less does not have a cursor to begin with.
It would need one to navigate to start and end of the text to select.
less is just not the right tool for character-level navigation.

Tabs already expanded

You can use the key shift and the mouse to make a selection; This is handled by the terminal, not by less. But the terminal does not know how spaces and tabs where arranged - less does the interpretation of tabs internally, and writes only normal " " characters to the screen.

There are tools like screen, tmux and byobu, which can do lots of impressive things in this area. I did not check, but I assume that these terminal multiplexers do not have a way around that - being terminals, in the end - and will behave the same.

Use vim

If you are showing a file in less, there is a nice solution:
Press the key v in less to open the current file in vim - asuming your $EDITOR etc. is set up for vim.

This does not work when showing stdin from a pipeline or so, although there are workarounds.

Mouse scrolling, at least

But you can at least scroll with the mouse wheel:

That seems even to be enabled by default, but the mouse wheel events get suppressed by a different option.

For a quick test, try:

LESS=-r man less

The option -X (--no-init) blocks scrolling - check what your environment variable LESS contains:

$ echo $LESS

The option -q (--quiet, --silent) also causes trouble, according to SU: How to make mouse wheel scroll the less pager using bash and gnome-terminal?