Is it generally a given that Burton's two films and Shoe-mockers (doesn't deserve the effort to spell it right) are sort of in the same continuity?
In theory, yes, they were meant to be. They share Michael Gough's Alfred and Pat Hingle's Commissioner Gordon. And Forever
had an indirect reference to Catwoman in dialogue. Burton was originally going to direct the third film, and he did produce it. But when Schumacher took over, he did a wholesale revamp on the series, so it has a different feel and sensibility despite being nominally in continuity.
At the time, ...Forever was viewed as the third Batman film, the sequel to Batman and Batman Returns. But I think that had it been made today, then it would have been described as a reboot - different actor, different director, different tone, different look for Gotham, different soundtrack, different Batmobile, different costume, etc.