Tobias Peitzsh help us identifying the root of the issue:

In method 
@Override protected Boolean run() throws Exception { 
return getContext().get(FilePath.class).child(file).exists(); 
} 
child method is called. Child method creates a new FilePath object. While creating the object, method resolvePathIfRelative is called. This method replaces the relative path with "base.remote+'\\'+rel.replace('/','
')". So "" is interpreted as "." (current directory). 
--> If the current directory exists, fileExists returns true otherwise false. 
Please see the following script: 
node('GF_SI-Z0TTN_SWBUILD'){ 
step([$class: 'WsCleanup']) 
echo fileExists("").toString() // current workspace does not exist 
bat "echo test>test.txt" 
echo fileExists("").toString() // current workspace exists 
} 
Output is 
false 
true 
That is the issue why I got true / false on different machines. The script was called on different machines before. On these machines I got true for the others I got false.

The Java behavior of new File("").exists() is as expected: 
System.out.println(new File("").exists()); // --> always false 
System.out.println(new File(".").exists()); // --> always true

So at least a hint for fileExists() is needed to note the unexpected behavior.

My use case is: A string parameter specifies a file to be used during the build (e.g. special parameters or test to be executed). If the file exists the action shall be done. If the file does not exists the action shall not be done. So I wrote (simplified) 
if (fileExists(env.FileParameter)){ 
// do special action 
} 
else{ 
// do nothing 
} 
So if FileParameter is empty / "" the action is executed although the file does not exist. 
Using if (env.FileParameter.trim() !="" && fileExists(env.FileParameter)) is not obvious.

PR up at https://github.com/jenkinsci/workflow-basic-steps-plugin/pull/56 that adds a log message when fileExists is called with an empty string informing the user that the current directory is checked instead.

Code changed in jenkins
User: Andrew Bayer
Path:
pom.xml
src/main/java/org/jenkinsci/plugins/workflow/steps/FileExistsStep.java
src/main/resources/org/jenkinsci/plugins/workflow/steps/Messages.properties
src/test/java/org/jenkinsci/plugins/workflow/steps/CoreWrapperStepTest.java
src/test/java/org/jenkinsci/plugins/workflow/steps/FileExistsStepTest.java
http://jenkins-ci.org/commit/workflow-basic-steps-plugin/35c716699dc87bf0a1188e91c9c7fa9c61603ea2
Log:
[FIXED JENKINS-48138] Log a warning when fileExists called with empty string

That warning will be in the next release of workflow-basic-steps (2.7, I believe).