This answer is written keeping your current context in mind and my experience working in India.
What I want to know is if I abscond this organisation, would they be able to take any serious actions?
I am assuming you have already completed the paperwork and signed the offer letter and any other papers. So, absconding is never a good idea. It could be very well argued if the company will take any legal actions, but this is definitely not something you should risk doing.
Are you currently under probation? You should be able to check if that's the case in from your offer letter, or from the HR. Probation is applicable both ways and generally the release duration under probation is much lesser (days) compared to what it is after job confirmation.
Can they get me blacklisted with other organisations and create a problem in future?
There's no official/publicly known data about this, and it also depends on whether you were hired through internal or external recruiter, but you should never risk it. You certainly stand a strong chance to get blacklisted from the current organisation and it's subsidiaries.
My new company is okay with me absconding this MNC, and I want to join them in just a days gap. Would this create any problems for me in the future?
This is a big red flag in my opinion. An employer which encourages its potential employee to take such action is not a reliable one in my opinion. Any organisation employing/suggesting such actions stands a chance to turn their back against you in future.
Evaluate how lucrative the offer from the mid sized company is in the following following terms in the given order:
Domain and quality of work.
Skill and expertise level of the current team members.
Alignment with your career progression and future goals.
Pay, perks and general employee benefits.
If and only if the above points check in the listed order, you should think about leaving the MNC, and even if so, do it by communicating to proper channels and serving the applicable notice period.
The new company is worthy of joining (and you too can check your desirability by the new organisation), if they respect your current situation and handle it professionally by allowing required time between transition.
You mentioned that the notice period is 60 days when under probation. You haven't started getting involved in the work yet, you can try and ask for an expedited release. But it would make for an awkward conversation.
If the MNC don't agree to expedite the release, you may end you burning bridges there, while also standing a chance to lose offer from the new company. Give this a good though before making any move.
Also, the way how MNCs in India operates and follow proper protocols, you may end up serving 60 days notice. Just make sure you don't end up in a situation where you don't have any option in hand.