After doing a lot of searching for an answer to the same question myself, I finally came up with an answer. The short answer is:
echo 80 | sudo tee /sys/devices/platform/sony-laptop/battery_care_limiter
The long answer is, if you look at the available options in the Windows 7 version of the Sony Battery Care application you will see that the only available options are to set the charging limit to 50, 80, or 100. The other value that works is 0, which from what I can tell turns off the battery care function and allows the battery to charge to 100% or as much as it can hold.
So, using the code example above, echo <number>
where "number" = 0, 50, 80, or 100 will give you the effect you're after.
Note: This works on my Sony Vaio Model VPCEB15FM running Ubuntu 18.04, your mileage may differ.
Additional Info:
Under the folder /sys/devices/platform/sony-laptop/ there are some other options as well. What I noticed is that several of these files have similar data/parameters as the Windows 7 version of the Sony Battery Care application. It looks like these can be used to do similar tweaks. I haven't looked at them all, but here is a short list with some notes.
battery_care_health
Always 50. For some reason can't be changed even after doing a chmod.
However, for some reason, it did reset to 0 after the battery completely discharged.
battery_care_limiter
0 - starts charging immediately.
50, 80, 100 seem to be only valid settings
thermal_profiles
displays "balanced performance" which is the same choices in Windows Vaio Care
thermal_control
currently "balanced" which is the same setting in Windows Vaio Care
lid_resume_S3
lid_resume_S4
lid_resume_S5
all these are 0. probably have something to do with init/load/boot states
driver_override
currently "(null)"
panel_id
currently "7"
modalias
currently "platform:sony-laptop"
uevent
DRIVER=sony-laptop
MODALIAS=platform:sony-laptop
power
folder with some more settings. Can't tell what they do.
Some can't even be read
contains these files:
async runtime_active_kids runtime_status
autosuspend_delay_ms runtime_active_time runtime_suspended_time
control runtime_enabled runtime_usage
driver
folder with what looks like the driver info
more folders:
bind module sony-laptop uevent unbind
touchpad
currently "1" - best guess is that it enables the touchpad
subsystem
more folders:
devices drivers drivers_autoprobe drivers_probe uevent