p.1 #1 · Are the Leica M-P 240 and M-E Typ 240 basically the same camera?
Apologies, but a quick Google search comparing the MP240 and newer ME240 only yielded one Leica Forum thread and that was originally intended for the M 240 (1GB buffer).
Are these cameras functionally the same, but just differ in that the M-P has black/silver paint while the M-E only comes in "anthracite"?
p.1 #3 · Are the Leica M-P 240 and M-E Typ 240 basically the same camera?
Slight differences: The M-E 240 has limitations compared to the M-P 240 when it comes to accessory handgrip #14495. You can attach it to the M-E 240 body, but all (or most?) electronic functions like GPS will remain disabled. This information can only be found in a little addendum sheet Leica added to the camera manual (the original manual which comes with the camera mentions #14495 as suitable handgrip option which is false). If you decide to get a handgrip for the M-E 240, you might be better off with the cheaper version without electronic features, #14496.
The M-P 240 also has sapphire glass LCD protection whereas the M-E 240 uses Gorilla glass. But technically they are otherwise the same - both also use the same 2 GB buffer being a big benefit over the regular M 240. Both the M-E 240 and M-P 240 are made of magnesium alloy chassis; top and bottom plates are made of brass. The M-E 240 uses anthracite paint whereas the M-P 240 came in different paint versions.
I have the M-E 240 and really like this camera! Only 750 pieces of it were made. This model is a bit of a hidden gem since much less of it can be found online than for the M-P version. It does not officially count as collector item (the E stands for entry - in this case only due to the minor differences pointed out above), Leica likely used in-stock body items for the M-E model while already producing the M 10 at the time in 2019.
p.1 #4 · Are the Leica M-P 240 and M-E Typ 240 basically the same camera?
I suspect the M-E 240 was likely the last entry camera line of Leica made! This is after Leica removed "cheaper" brand items like the Summarit lens line with f/2.4 and f/2.5 lenses, and the TL and later also CL camera lines. A Leica M-E 10 is likely never going to be released after the M 11 was announced.