The short answer is that this is how he referred to himself. He was born Tyge Otteson Brahe but at the age of 15 (1561) changed 'Tyge' to Latinized 'Tycho', see Redd's biography of Tycho and Thoren's book Lord of Uraniborg. Johannes Müller, a well-known German astronomer, wrote under 'Regio Monte', which Melanchton, an educational authority at the Copenhagen university where Tycho studied, Latinized into 'Regiomontanus' in 1534. Mononyms were common among ancient Latin authors (Pliny, Vitruvius), especially poets (Ovid, Horace, Virgil), although those were derived from middle names. Tycho was very much into Latin poetry, and composed himself, see Tycho Brahe in Scandinavian Scholarship by Christianson:
"Zeeberg began by placing Tycho in the first rank of Danish Latin poets and examined
chemical images in one of his major poems (Zeeberg 1987). Next, he identified
Ovid as Tycho's model for a poetic persona and style and went on to analyse a
poem of 1584, dedicated to Tycho's friend, Erik Lange, which used a conception of
love (amor) derived from Marsilio Ficino (Zeeberg 1988, see also Zeeberg 1991b).
The poem was replete with chemical imagery and aimed to turn Lange's obsessive
love of alchemical goldmaking into a true love focused on the Paracelsian chemistry
[...] In his first Urania poem
of 1573, Tycho pictured himself as Urania's humble pupil and described astronomy
as an eternal and divine discipline, comparing it to the ignoble, earthbound pursuits
of most noblemen. Tycho rejected the customary noble lifestyle of "ambition, moneymaking,
ignorance, and luxury", and this rejection, according to Zeeberg, "had an
influence on the colossal self-confidence and superior attitude towards norms and
rules which became so characteristic of him". Later, when Tycho became the noble
lord of Uraniborg, he decided that noble rank and intellectual achievement were not
irreconcilable and began to ascribe the worldly lifestyle, not to noblemen, but to "the uncomprehending mob" (Zeeberg 1991a, Zeeberg 199lb). Tycho's friendship
poems of 1584-85 adapted the Renaissance theme of translatio musarum, depicting
Urania as refugee in many lands before she finally found her home on the island
Royals and nobles, which Tycho also was, were often known by first names, like his benefactors, Frederik and later Rudolf, and Tycho had many siblings who shared his family name. Many Italian Renaissance figures, mostly poets and artists (Dante, Michelangelo), but some scientists too (Leonardo, Galileo) used first name mononyms, possibly in imitation of revered Greek ancients (Plato, Euclid, Archimedes, Hipparchus, etc.). Any combination of those reasons might have influenced Tycho's choice.
More is known about the naming practice in Italy, as in the cases of Galileo and Leonardo, see Why Do We Call Galileo Galilei by His First Name?:
"Galileo went on to become one of the most recognized names in scientific history. But why do we call him by his first name only?
Because that’s how he referred to himself. At the time of Galileo’s birth in 1564, surnames were optional in Italy. In daily interactions, an Italian would use the name his parents gave him at birth—what we’d now call a first name — and, if further clarification were required, add on his father’s name (like di Antonio, or “son of Antonio”), his birthplace (Romano, or “from Rome”), his occupation (Panettiere, meaning “baker”), or a traditional family surname (if one existed, like Galilei). The Italian astronomer’s name is unusually confusing because both Galileo and Galilei were surnames used by his family for generations. (An equivalent might be “William Williams.”) This was not a particularly common practice at the time. Moreover, the name Galileo itself, although not completely unique, was quite rare. This is part of the reason we continue to use his first name only — it’s unambiguous.
In Renaissance Italy, individuals didn’t even stick with the same second, or identifying, name throughout their lives. Many used their family surnames one day and place of birth the next, depending on the circumstances. Take Leonardo da Vinci. Because Vinci was a very small town, calling himself Leonardo from the town of Vinci left little room for confusion—unless, of course, he was in Vinci at the time. (Leonardo was a common name.) In that case, the artist would probably have called himself Leonardo di ser Piero da Vinci, making reference to his father. Once he became famous, he often signed his name simply “Leonardo.” Galileo referred to himself sometimes by first name only, sometimes as Galileo Galilei, and sometimes as Galileo Galilei Linceo (a nod to his alliance with a progressive group of scientists, which served, in part, as a kind of honorific)."