Selecting eclipse.desktop icon doesn't do anything (visible to me), "Exec format error" from shell

Question

Trying to execute via desktop icon:

/usr/bin/eclipse $ ls -l eclipse
-rwxrwxrwx 1 geri geri 61176 Sep 16 19:12 eclipse

Right-click on desktop → Create launcher... creates:

~/Desktop# ls -l eclipse.desktop 
-rw-rw-r-- 1 geri geri 195 Dez 12 13:37 eclipse.desktop

~/Desktop# cat eclipse.desktop 
[Desktop Entry]
Version=1.0
Type=Application
Name=eclipse
Comment=
Exec=/usr/bin/eclipse/eclipse
Icon=/home/geri/Pictures/logo-eclipse.png
Path=/usr/bin/eclipse
Terminal=false
StartupNotify=true

Double-clicking on desktop icon:

Selecting Launch anyway does nothing (visible to me).

Selecting Mark Executable gives:

~/Desktop# ls -l eclipse.desktop 
-rwxrwxr-x 1 geri geri 195 Dez 12 13:37 eclipse.desktop

Double-clicking on desktop icon now does nothing (visible to me).

I added my user (geri) to the root group and tried ~/Desktop/eclipse.desktop and /usr/bin/eclipse/eclipse with owner/group root/root as well, to no avail.

Checked:

$ java --version
openjdk 11.0.4 2019-07-16
OpenJDK Runtime Environment (build 11.0.4+11-post-Ubuntu-1ubuntu218.04.3)
OpenJDK Server VM (build 11.0.4+11-post-Ubuntu-1ubuntu218.04.3, mixed mode, sharing)

$ echo $JAVA_HOME
/usr/lib/jvm/java-11-openjdk-i386

Adding the following to eclipse.ini (as per Is it necessary fix eclipse.desktop to use Eclipse launcher in Ubuntu 16.04?) didn't help:

--launcher.GTK_version
2

Adding the following to eclipse.ini (as per How do I create a .desktop file to launch eclipse?) didn't help (I'm not that far, however, that something [an error] is displayed.):

-vm
/usr/lib/jvm/java-11-openjdk-i386/java/bin

Trying from shell:

/usr/bin/eclipse $ ./eclipse
bash: ./eclipse: cannot execute binary file: Exec format error

/usr/bin/eclipse $ readelf -l eclipse
Elf file type is EXEC (Executable file)
Entry point 0x400fe0
There are 9 program headers, starting at offset 64

Program Headers:
  ...

Why the error if it can be read as ELF file?

/var/log $ grep "eclipse" *.log doesn't show anything.

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N.B.: This really s***s. Installing Eclipse on Windows from scratch is a matter of half an hour, at most, including download and installation of JDK. I'm fiddling around with this and walking through manuals/forums for two half days now.

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UPDATE

$ uname -vmpi
#40~18.04.1-Ubuntu SMP Thu Nov 14 12:06:25 UTC 2019 i686 i686 i686

/usr/bin/eclipse $ file ./eclipse
./eclipse: ELF 64-bit LSB executable, x86-64, version 1 (SYSV), \
dynamically linked, interpreter /lib, for GNU/Linux 2.6.32,  \
BuildID[sha1]=b621fbc20e80d7ac375ece11a5775fc6686b71c7, with debug_info, not stripped