The Canadian Trotting Association and USTA require that horses be registered before turning one. I believe this is the registration site for the Canadian organization Changes To Canadian Registration Requirements For OSS-Nominated Horses | Standardbred Canada
You may be able to register in another way. Even if you were able to register it would require at least a freeze brand and a DNA test I.e. "Canadian registration through Standardbred Canada requires that the foal be freeze-branded and DNA tested by an Identification Technician before registration can be completed."
In most horses registration does increase value but for standardbreds because the value is based in lines or track success if you have no intention of breeding her or racing her there are no great benefits. Which means that in the wider world it does not have much value. No one buys a pleasure standardbred because its registered with the USTA. Most horses that are bought by people who care about registration in the standardbred world are bought based on lines or race record of which would require registration. I have not heard of many non-registered standardbreds unless they were bred by the Amish or were crosses.
I ask about family names because parent lines can have some fun names like Mr. Telephone man, Beach towel or precious bunny.