Hey Botond and welcome to the community 
Seems like overlooked the examples in the documentation, there’s one for your exact use case (the second last one):
# Sets date and time to date and time from datetime object (current date and time in this example)
- service: input_datetime.set_datetime
entity_id: input_datetime.date_and_time
data_template:
datetime: "{{ now().strftime('%Y-%m-%d %H:%M:%S') }}"
Hi fedot, thanks for your reply. I certainly tried it on the base of documentation, but did not work.
Expected a value of type {action,navigation_path,url_path,service,service_data} | undefined for tap_action.entity_id but received "input_datetime.jacu".
I tried this:
type: picture
image: 'https://demo.home-assistant.io/stub_config/t-shirt-promo.png'
tap_action:
action: call-service
service: input_datetime.set_datetime
entity_id: input_datetime.jacu
data_template:
datetime: '{{ now().strftime(''%Y-%m-%d %H:%M:%S'') }}'
Any help would be appreciated.
Botond
Ah yes, the frontend sometimes requires an entity_id even if you don’t need it.
Try the one you’ve just posted, add entity_id: sun.sun (eg. below “type”)
Hi fedot,
I added sun entity below type, but it does not work either. My code is:
type: picture
entity_id: sun.sun
image: 'https://demo.home-assistant.io/stub_config/t-shirt-promo.png'
tap_action:
action: call-service
service: input_datetime.set_datetime
entity_id: input_datetime.jacu
data_template:
datetime: '{{ now().strftime(''%Y-%m-%d %H:%M:%S'') }}'
Error message right below the code editor:
Expected a value of type undefined for entity_id but received "sun.sun".
Error message when clicked on the image:
Failed to call service input_datetime.set_datetime, must contain at least one of entity_id, area_id.
Oh, I’m sorry, it’s a picture card, not a picture entity card. Try this please:
type: picture
image: 'https://demo.home-assistant.io/stub_config/t-shirt-promo.png'
tap_action:
action: call-service
service: input_datetime.set_datetime
data_template:
datetime: "{{ now().strftime('%Y-%m-%d %H:%M:%S') }}"
entity_id: input_datetime.jacu
Thank you, fedot!
Unfortunately still does not work:
Expected a value of type {action,navigation_path,url_path,service,service_data} | undefined for tap_action.data_template but received {"datetime":"{{ now().strftime('%Y-%m-%d %H:%M:%S') }}","entity_id":"input_datetime.jacu"}.
Yes, you’re right.
I just looked it up again, you can’t template in Lovelace.
Create a new script that calls the service with the data_template and then call the service from the tap_action. Sorry for the inconvenience 
Dear fedot, thanks for your effort! 
An alternate way to doing this:
datetime: "{{ now().strftime('%Y-%m-%d %H:%M:%S') }}"
is this:
datetime: "{{ now().timestamp() | timestamp_local }}"
or even this:
datetime: "{{ (now() | string)[:19] }}"
Dear Taras,
unfortunately this does not work either.
Error message is the same as above.
Thank you! 
Yes, he’s talking about the template you have to use in your script. You can’t template in Lovelace, as I said.
Need a few hints for the script? Can write it for you if you want to
Dear Botond,
The templates I posted do work and the evidence is in the screenshot.
What does not work is attempting to use any template with tap_action. As fedot mentioned, it doesn’t support templating.
Simply use the template in a script then have tap_action call the script. It’s easy.
Hello guys,
thanks for info, it works with script!
Good to know that templating does not work in Lovelace.
Have a nice day!
That’s a generalization and it doesn’t reflect the facts. The ability to support templating is not determined by Lovelace but by the card used.
The current set of standard cards don’t support templating for tap_action. However, there are various custom cards that do support templating. For example, the Auto-Entities card.
I don’t know how it works, but it worked!! Thank you, search took very long.
Found this old thread.
I am trying to do the same thing but actually with a script.
Somehow it is not working.
alias: New Script
sequence:
- service: input_datetime.set_datetime
data:
datetime: "{{ now().timestamp() | timestamp_local }}"
target:
entity_id: input_datetime.end_datetime
mode: single
alias: New Script
sequence:
- service: input_datetime.set_datetime
data:
timestamp: "{{ now().timestamp() }}"
target:
entity_id: input_datetime.end_datetime
mode: single
Thank you @123 , but that script also cannot be saved. 
It does not show an error but it is not stored and hence cannot be run (not even in the UI editor).
So half a step closer but not there yet.
And why does it not throw an error for your code?
Mine is from the docs. Are the docs outdated or is my syntax wrong? I cannot see any problem.
I was able to create/save/store both examples (the one from the documentation and mine). I don’t know why you’re encountering a problem saving them.
Checking at the moment if it is a local issue with my supervisor setup (corrupted something) or if this is a problem with me using packages:
When I create the script as yaml in packages, it works.
When I create it manually in scripts.yaml or save it via UI(which is written to scripts.yaml) neither show in the UI.
So the scripts.yaml entries are not shown in the UI. So maybe incompatibility with packages? All scripts in packages worj flawlessly (including these datetimes).