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I have submitted an article for a journal in which it was stated that the review time was from 80 to 120 days, that is approximately 4 months at most. The submission of my article was approximately 13 months ago and I did not get any answer in the allotted time they mentioned. I sent an email to the editor after 8 months and I was told that the review process has suffered a delay, but that they soon will fix that issue.

I waited until September and sent another email and again I got another reply of an apology and nothing more. I believe that it has passed too much time, until now it is almost like one year, so I do not what to do.

Would there be any problem if I submit my article to another journal or should I send another email to the editor of this journal? Or rather just wait until I get a response?

It seems like a long time by any standard. Contact them. You can give them a date by which you will formally withdraw your article from consideration. Maybe a couple of weeks. If they don't jump you can submit elsewhere without worry.

The final nudge is just a courtesy. You could actually just inform them that you are withdrawing for submission elsewhere. But if the journal is reputable it might be worth the courtesy.

I agree with @Buffy's answer. A polite letter to the editors asking for clarification would be good. If you or a colleague know anyone on the editorial board - contact them. I had a case where a paper was sitting idle for close to a year after an accept with minor revisions because of a miscommunication between the reviewers and the editor in charge.

Another point to consider: perhaps I am being a bit paranoid, but if your paper was sitting there for a very long time and has not been published, it may be a good idea to have a version of it on ArXiv or some other relevant open repository. This serves the purpose of timestamping your publication.

There are (thankfully rare) horror stories of unscrupulous reviewers purposely delaying decisions in order to get the results themselves. If your review is taking so long and results in your field take a long time to come by (say, experiments need to be run), this may be a cause for concern.

Also seriously consider whether this journal is worth submitting to in the future. Journals should not be rewarded for this kind of behavior.

Write to the editor with pointed questions. How many reviewers have been invited? How many agreed/declined? When are the review due dates?

Without knowing the answers to these questions, deciding whether to wait or to withdraw and resubmit is just a crapshoot. With the answers, it's possible to make a much more informed decision about whether the reviewers are likely to finish their reviews.

If the editors refuse to answer, you can still guess if the delay is because of them or because of the reviewers based on how long it takes to answer your question. If they take a long time to answer, I'd guess that the delay is because of them, in which case I'd be more inclined to withdraw and submit elsewhere.

Three times in my (pretty long) career editors took an outrageously long time (more than a year) to decide on a submission. I was confident that the papers were correct and appropriate for those journals, so did not want to withdraw them and resubmit elsewhere. Eventually my frequent mail to the editors (snail and later e-) led to acceptance in each case. In at least one of them I think the editor gave up on nagging the referees and checked the paper herself.