Is there an issue if someone drinks directory out of a plastic bottle after eating meat and then drinking out of the same bottle after having a milky meal?
Very good point, yes there is. During the milky meal he should not drink from the bottle, since there is residue from the meat meal in and on the bottle. After the meal however he may drink from it, however it is preferable to wash it out first.
Y:D 88-2, although the above is talking about two people eating at the same time, it would apply also to one person eating meat while drinking from the cup, and then drinking from the cup while he eats milk. however this is only when they are eating together, but afterwards is different, see Badei Hashulchan 88-28, Zerah Shmuel Ftnt 92, Ohel Yackov 89-18.