'Break away' implies more violence than 'separate'.
Separating can happen by mutual consent or intentional design. It can be reversible. It sounds technical and formal.
Breaking away sounds more permanent and destructive. It can imply more emotion. It can imply a one-sided separation or unintended structural failure.
But it isn't idiomatic for the end of romantic relationships. That's usually 'breaking up' instead of 'breaking away'.
If a person or group were described as breaking away from another person or group, it would imply something political or violent to me.
Both expressions are more common in formal contexts like narrative, technical reports, etc. They are both less common in informal dialog.
Neither expression would be used for a dog or child as in the OP's first example.
"Don't let your kid wander away." "Keep you kid with you." "Keep your dog on a leash." Would all be more idiomatic.
Separated is only idiomatic for two individuals as in the OP's second example if it involves a romantic relationship or a formal narrative. 'Broke away' would not be idiomatic in that situation.
Examples that are idiomatic:
The boosters separated from the rocket at 30,000 feet. (A formal context.)
The wing broke away from the fuselage. (This is a formal, narrative context. In informal dialog something like "The wing tore off" or "The wing came off" would be more common.)
Paul broke away from Doug and ran. (Implies Doug had physically restrained him or that they were involved in combat or at least a confrontation. This is narration, not dialog. Dialog covering this situation might be "Then I threw him off me and ran!" or "Then I turned away and ran!")
The institutionalists broke away from the conservatives four years ago. (Idiomatic for political factions splitting from one another.)
Paul and I separated at four o'clock and I have not heard from him since. (Separated here could mean they ended a romantic relationship or it could be a formal narrative context, as in a police report or military debriefing. It would not be common in informal dialog. For informal dialog it would be something more like "I haven't seen Paul since about four." Or "Paul and I left at four and I haven't seen him since.")
Paul and I broke up at four o'clock and I have not heard from him since. (This is the end of a romantic relationship or a formal meeting depending on the context.)
The meeting broke up at four o'clock and I haven't heard from Paul since.
We got seperated in traffic. (Seperated is idiomatic in informal dialog for people who are traveling together rather than leaving separately from the same place.)
The group separated and went home. (Works in narration, would not be idiomatic in informal dialog. The dialog equivalent would be something like "Then we all went home.")
Team four will separate from the column at four o'clock. (A formal context.)
Team four will break away from the column at four o'clock.