Prince Albert was born at Schloss Rosenau near Coburg on 26 Aug 1819, the son of Ernest III, Duke of Saxe-Coburg-Saalfeld, and Louise of Saxe-Gotha-Altenburg. He was called Francis Albert Augustus Charles Emmanuel. Queen Victoria came to the throne on 20 June 1837, aged 18. Albert was introduced to her and they married on 10 Feb 1840. He was HRH Prince Albert, Duke of Saxony and did not have the title Prince Consort until 25 June 1857. A letter from the Adjutant-General dated 26 Aug 1842 informed the CO of the 13th that 'Her Majesty has been graciously pleased to approve of that Regiment assuming the title of the 13th or PRINCE ALBERT'S REGIMENT OF LIGHT INFANTRY; and its facings being changed from yellow to blue.' The wording implies that there was an appeal to the Queen for Prince Albert to put his name to the regiment but the History of the 13th does not mention this. There is also no indication that Prince Albert asked to be associated with the 13th who were being feted as the heroes of the hour.
It would be reasonable to assume that he was Colonel-in-Chief of the 13th Light Infantry but I can find no confirmation of this. He was Colonel of five regiments but Colonel-in-Chief of none:
8 Feb 1840 Appointed Field Marshal (2 days before his wedding to Queen Victoria)
Regimental Details | Colonels in Chief