Executing Trace on an expression reveals what is actually happening:
Trace[Remove@x; x = 1]
(*{Remove[Removed[x]];Removed[x]=1,{Remove[Removed[x]],Null},{Removed[x]=1,1},1}*)
"the Wolfram Language always reads in a complete input expression, and interprets the names in it, before it executes any part of the expression." (see: https://reference.wolfram.com/language/tutorial/SettingUpWolframLanguagePackages.html)
The whole line is one expression with the head CompoundExpression. The variable x gets replaced with Removed["x"] throughout that expression. Then it seems that Removed["x"] is assigned the value 1 (can someone clarify what is happening here?).
However, when you write it on two different lines, it is interpreted as two separate expressions and hence you don't face this problem.