Is it allowed to say that "this person saved my life"?
And also "If it was not for him I would have been dead"?
Also "We won the match because of him"?
Are these statements contradicting to Tawhid because ALLAH'S name has to be used in them instead of People?
It says words to this effect in the Qur'an:
مِنْ أَجْلِ ذَٰلِكَ كَتَبْنَا عَلَىٰ بَنِي إِسْرَائِيلَ أَنَّهُ مَن قَتَلَ نَفْسًا بِغَيْرِ نَفْسٍ أَوْ فَسَادٍ فِي الْأَرْضِ فَكَأَنَّمَا قَتَلَ النَّاسَ جَمِيعًا وَمَنْ أَحْيَاهَا فَكَأَنَّمَا أَحْيَا النَّاسَ جَمِيعًا وَلَقَدْ جَاءَتْهُمْ رُسُلُنَا بِالْبَيِّنَاتِ ثُمَّ إِنَّ كَثِيرًا مِّنْهُم بَعْدَ ذَٰلِكَ فِي الْأَرْضِ لَمُسْرِفُونَ
Because of that, We decreed upon the Children of Israel that whoever kills a soul unless for a soul or for corruption [done] in the land - it is as if he had slain mankind entirely. And whoever saves one - it is as if he had saved mankind entirely. And our messengers had certainly come to them with clear proofs. Then indeed many of them, [even] after that, throughout the land, were transgressors.
The relevant Arabic word is أحيا (ʾaḥyā) which is usually translated to "save" in this context. It therefore seems reasonable to say someone "saved" someone else's life. And it's not hard to find examples of scholars using this language (e.g. save a Muslim's life IslamWeb).
More generally, nothing happens except by the will of Allah (Qur'an 6:17, 9:51, 57:22). Or as Islam Q&A put it: "Nothing happens in His Dominion except that which He wills". Nevertheless, verses such as
And that there is not for man except that [good] for which he strives
indicate that humans are capable of causing things to happen (albeit constrained by Allah's will). So it's normal to say "I did this", "you did this", "Prophet Muhammad did this", etc., in everyday conversation.
It is more appropriate to thank Allah first, and then others. This is more appropriate in terms of Allah's control over the situation, over the one who helped you, over causing the situation in the first place out of His wisdom, and so on.
However, it is not shirk, unless you believe that it was another person who saved your life to the exclusion of divine help. Note that words are a part of faith (and in some circumstances can nullify faith), however actions of the body (physical actions) and actions of the heart (tawakkul/trust in Allah, love of Allah, etc) are also a part of faith. As a Muslim, no doubt within such situations, you would be praying and making dua to Allah alone within moments of such events, so the actions of your body and heart would be a proof that you thanked another only as a secondary helper, with the primary help being from Allah.
It is also important to realize that in Islam, we have words of remembrance to say during all occasions, and this constantly reinforces the reality of Allah's help and guidance and our need for that. The best therefore is to keep Allah at the forefront of all acts of thankfulness, and to thank others afterward.
And Allah knows best.