Maybe in Canada (don't know the details of their DUI statutes), but fortunately not in California. Here in California there is a separate part of the vehicle code specifically dealing with BUI:

21200.5. Notwithstanding Section 21200, it is unlawful for any
person to ride a bicycle upon a highway while under the influence of
an alcoholic beverage or any drug, or under the combined influence of
an alcoholic beverage and any drug. Any person arrested for a
violation of this section may request to have a chemical test made of
the person's blood, breath, or urine for the purpose of determining
the alcoholic or drug content of that person's blood pursuant to
Section 23612, and, if so requested, the arresting officer shall have
the test performed. A conviction of a violation of this section
shall be punished by a fine of not more than two hundred fifty
dollars ($250). Violations of this section are subject to Section
13202.5.

Note that the fine specified is much lower than for DUI and there is no provision for jail time,
loss of driver's license, or other penalties.
The reference to Section 13202.5 is for juveniles between the ages of 13 and 21 who do not yet have a driver's license.
In that case the court can delay their eligibility to get a driver's license if they are guilty of BUI.

Seems more rational than the situation in the OP's case and other locations where there is no distinction between BUI and DUI.